Constructive Analysis of Semicircular Elements. From Orthogonal Projections to Semicircular Elements 9781032448336, 9781032449814, 9781003374817

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Constructive Analysis of Semicircular Elements This book not only examines the constructions and free-probabilistic properties of semicircular elements, as defined within the text, but also considers certain Banach-space operators acting on these semicircular elements and shows how they deform (i.e., preserve-or-distort) the semicircular law induced by orthogonal projections. Features • Suitable for graduate students and professional researchers in operator theory and/or analysis • Numerous applications in related scientific fields and areas

Monographs and Research Notes in Mathematics Series Editors, John A. Burns, Thomas J. Tucker, Miklos Bona, and Michael Ruzhansky This series is designed to capture new developments and summarize what is known over the entire field of mathematics, both pure and applied. It will include a broad range of monographs and research notes on current and developing topics that will appeal to academics, graduate students, and practitioners. Interdisciplinary books appealing not only to the mathematical community, but also to engineers, physicists, and computer scientists are encouraged. This series will maintain the highest editorial standards, publishing well-developed monographs as well as research notes on new topics that are final, but not yet refined into a formal monograph. The notes are meant to be a rapid means of publication for current material where the style of exposition reflects a developing topic. Double Sequence Spaces and Four-Dimensional Matrices Feyzi Ba¸sar and Medine Ye¸silkayagil Sava¸scı Banach-Space Operators on C*-Probability Spaces Generated by Multi Semicircular Elements Ilwoo Cho Applications of Homogenization Theory to the Study of Mineralized Tissue Robert P. Gilbert, Ana Vasilic, Sandra Klinge, Alex Panchenko, and Klaus Hackl Constructive Analysis of Semicircular Elements: From Orthogonal Projections to Semicircular Elements Ilwoo Cho Inverse Scattering Problems and Their Applications to Nonlinear Integrable Equations, Second Edition Pham Loi Vu

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Constructive Analysis of Semicircular Elements From Orthogonal Projections to Semicircular Elements

Ilwoo Cho

St. Ambrose University, Davenport, Iowa

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First Edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN

© 2023 Ilwoo Cho

CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-44833-6 (hbk) ISBN: 978-1-032-44981-4 (pbk) ISBN: 978-1-003-37481-7 (ebk) DOI: 10.1201/9781003374817 Typeset in Minion by codeMantra

Contents Author

ix

CHAPTER 1

Introduction

1

1.1

FREE PROBABILITY

3

1.2

SEMICIRCULAR ELEMENTS

4

1.3

SOME FREE RANDOM VARIABLES INDUCED BY SEMICIRCULAR ELEMENTS

5

■

1.3.1 1.3.2 1.4

Circular Elements Free Poisson Elements

11

JOINT FREE MOMENTS OF MULTI SEMICIRCULAR ELEMENTS

15

REFERENCES

CHAPTER 2

■

5

30

Semicircular Elements Induced by p-Adic Number Fields Qp

33

2.1

ANALYSIS OF THE p-ADIC NUMBER FIELDS Qp

35

2.2

STATISTICAL MODELS INDUCED BY Qp

41

2.3

REPRESENTATIONS OF (Mp , φp )

44

2.4

MEASURE-THEORETIC MODELS ON {Mp }p∈P

46

2.5

C ∗ -SUBALGEBRAS Sp OF (Mp , φp,j )

49

2.6

STATISTICAL MODELS FOR BANACH ∗-ALGEBRAS LSp (j)

51

ON THE FREE ADELIC FILTERIZATION LS

58

2.7.1 2.7.2

60

2.7

Weighted-Semicircular Elements Weighted-Semicircular Elements of LS

61 v

vi ■ Contents

2.7.3

The Adelic Semicircular Filterization LS

64

2.8

CIRCULAR ELEMENTS OF LS

67

2.9

FREE POISSON ELEMENTS OF LS INDUCED BY U ∪ U

68

2.10 BANACH-SPACE OPERATORS ACTING ON LS

72

2.11 Pi-SHIFT OPERATORS ON LS

79

REFERENCES

85

CHAPTER 3

3.1 3.2

Semicircular Elements Induced by Orthogonal Projections

87

SOME BANACH ∗-ALGEBRAS INDUCED BY PROJECTIONS

88

WEIGHTED-SEMICIRCULAR ELEMENTS INDUCED BY Q

94

■

3.3

FREE-DISTRIBUTIONAL DATA ON LQ

103

3.4

CIRCULAR ELEMENTS OF LQ

105

3.5

FREE POISSON ELEMENTS OF LQ INDUCED BY U ∪ U 106

3.6

CERTAIN ∗-ISOMORPHISMS ACTING ON LQ

110

3.7

GROUP-DYNAMICAL SYSTEM ON LQ

116

3.7.1 3.7.2 3.7.3

116

3.8

Discrete-Group Dynamical Systems The Group-Dynamical System (B, LQ , α) The Dynamical Semicircular ∗-Probability Space LQ

BANACH ∗-PROBABILITY SPACES LQ [N ]

REFERENCES

CHAPTER 4

■

Certain Banach-Space Operators on LQ [N]

118 125 132 137 141

4.1

THE INTEGER-SHIFT-OPERATOR ALGEBRA B ON LQ

142

4.2

RESTRICTED-SHIFT-OPERATOR ALGEBRAS BN ON LQ [N ]

149

4.3

FREE PROBABILITY ON LQ [N ] DEFORMED BY BN

155

4.4

CONDITIONAL-NILPOTENT-PROPERTY ON BN

159

4.5

DEFORMED CIRCULAR LAWS ON LQ [N ] BY BN

164

Contents ■ vii

DEFORMED FREE POISSON DISTRIBUTIONS ON LQ [N ] BY BN

168

4.7

THE SEMICIRCULAR ∞-FILTERIZATION LQ [∞]

170

4.8

MORE ABOUT LQ [∞]

174

4.6

REFERENCES

CHAPTER 5

■

Discussion

REFERENCES Index

179 183 184 187

Author Ilwoo Cho is currently a Professor at St. Ambrose University, Iowa. He earned a PhD in Mathematics at the University of Iowa in 2005 and a Master’s degree at Sungkyunkwan University in 1999. Dr. Cho’s research interests include free probability, operator algebra and theory, combinatorics, and groupoid dynamical systems.

ix

CHAPTER

1

Introduction

he main purposes of this monograph are (i) to construct a free semicircular family from fixed countably many mutually orthogonal projections, (ii) to consider a Banach ∗-probability space generated by the free semicircular family of (i), (iii) to characterize the free-distributional data on the Banach ∗-probability space of (ii), (iv) to study certain operators acting on the Banach ∗-probability space of (ii), and (v) to investigate how the operators of (iv) deform the original free probability of (iii). The most important main result is how to construct mutually free semicircular elements from fixed mutually orthogonal projections. To do that, we begin with a special case; we consider how to construct semicircular elements from an analysis of p-adic number fields Qp , for a prime p ∈ P, where P is the subset of the natural numbers N consisting of all primes. Motivated by the construction of the semicircular elements from these analyses, we study mutually free, |Z|-many semicircular elements constructed from arbitrary |Z|-many mutually orthogonal projections in a certain C ∗ -probability space, whose free distributions are determined by nonzero real numbers in R× = R \ {0}. Then, Banachspace operators acting on those semicircular elements are studied under canonical shifting processes on indices of semicircular elements. And then, consider the case where there are N -many orthogonal projections for N ∈ N∞ >1 = (N \ {1 }) ∪ {∞},

T

where ∞ = |N|, the countable infinity.

DOI: 10.1201/9781003374817-1

1

2 ■ Constructive Analysis of Semicircular Elements

The study of semicircular elements is one of the main topics not only in both commutative function theory (e.g., [1,2]) and noncommutative free probability theory (e.g., [20,21,29,30]) but also in various applied fields including quantum statistical physics (e.g., [5–7,8,10–12]). In free probability theory, the semicircular law, which is the free distribution of semicircular elements, is well-characterized analytically and combinatorially (e.g., [17,18,21,28–30]). In particular, it plays a key role in free-probabilistic operator algebra theory by the (free) central limit theorem(s), e.g., see Refs. [2,17,19,28–30], since it becomes a noncommutative operator-algebraic analog of the classical Gaussian (or the normal) distribution (in commutative function theory). From combinatorial approaches (e.g., [17,22,23]), the semicircular ∞ law is universally characterized by the Catalan numbers {ck }k=1 , where cn =

1 n+1

2n n

=

1 n+1

(2n)! n!(2n − n)!

=

(2n)! , n!(n + 1)!

for all n ∈ N0 = N ∪ {0 }. i.e., the semicircular law is characterized by the free-moment sequence, ∞

(ωn cn )n=1 = (0, c1 , 0, c2 , 0, c3 , ...) , where

(1.1) def

ωn =

1 0

if n is even if n is odd,

and cn are the Catalan numbers, for all n ∈ N. From our analysis on p-adic number fields Qp (e.g., [26,27]), one can construct semicircular elements, e.g., see Refs. [5,12]. By generalizing the constructions of Refs. [5,12], semicircular elements were constructed whenever there are |Z|-many orthogonal projections in a C ∗ -algebra (e.g., [6–8,10,11]), different from earlier works (e.g., [20,25,29,30]). In this new approach, the semicircular elements are understood as Banach-space operators acting on a given C ∗ -algebra, by regarding the C ∗ -algebra as a Banach space equipped with its C ∗ -norm (e.g., [13,14]). Independently, the joint free distributions of mutually free, multi semicircular elements were re-characterized in Ref. [9] combinatorially

Introduction ■ 3

and analytically. Especially, there, the analytic characterization provided natural estimations and asymptotic estimations of such joint free distributions, i.e., we have a tool to verify how our adjointable Banach-space operators deform the original free-distributional data. Here, we introduce-and-consider combinatorial characterization of Ref. [9] in Section 1.4. Announcement. In this monograph, each section has its own references, and they are indicated at the end of sections.

✥

1.1

FREE PROBABILITY

In general, free probability is the noncommutative operator-algebraic analog of classical measure theory (including probability theory), e.g., see Refs. [3,15–17,28–30]. It is not only a major field in operator algebra theory (e.g., [9,17,20–22,28,29]) but also an interesting application in other applied fields (e.g., [4–8,10–12,24,25]). In this chapter, the combinatorial approach of Refs. [17,22,23] is used. Without introducing detailed definitions, or combinatorial backgrounds, the (joint) free moments and (joint) free cumulants are computed to verify free distributions of operators; the free-probabilistic free product of algebras is used to study operator-algebraic freeness (i.e., operator-theoretic version of classical independence) condition. As usual, if A is a topological ∗-algebra (a C ∗ -algebra, or a von Neumann algebra, or a Banach ∗-algebra, etc.), and if φ : A → C is a bounded linear functional on A, then the pair (A, φ) is said to be a (noncommutative) topological ∗-probability space (a C ∗ -probability space, respectively, a W ∗ -probability space, respectively, a Banach ∗-probability space, etc.). It is a free-probabilistic counterpart of a measure space. Operators a ∈ A are called free random variables if we understand a as elements of (A, φ). Two free random variables are free in (A, φ), if all mixed free cumulants of them vanish (e.g., [17,22,23]: See pure-analytic definition of freeness in Refs. [20,21,28–30]). Let a1 , ..., as ∈ (A, φ) be free random variables, for s ∈ N. Then the free distributions of them are characterized by the joint free moments, n

∞

∪

n=1

∪

(i1 ,...,in

)∈{1,...,s}n

arikk

φ k=1

: r1 , ..., rn ∈ {1, ∗}

,

4 ■ Constructive Analysis of Semicircular Elements

or by the joint free cumulants, ∞

∪

n=1

{

∪

(i1 ,...,in )∈{1,...,s}n

(1.1.1)

( ) } kn ari11 , ..., airnn : r1 , ..., rn ∈ {1, ∗} ,

by the M¨ obius inversion of Ref. [17], where k• (...) is the free cumulant on A in terms of the linear functional φ (e.g., [17,22,23]).

1.2

SEMICIRCULAR ELEMENTS

Let (A, φ) be a topological ∗-probability space of a topological *-algebra A, and a bounded linear functional φ on A. A free random variable a ∈ (A, φ) is self-adjoint, if a is self-adjoint in A in the sense that: a ∗ = a in A, where a ∗ is the adjoint of a. Note that the free distribution of a self-adjoint free random variable a is fully characterized by ∞

The free moment sequence (φ (an ))n=1 , and by

(1.2.1) 

∞



The free cumulant sequence kn a, a, ..., a ntimes

,

n=1

under the M¨ obius inversion of Ref. [17], by Eq. (1.1.1). Definition 1.1: A self-adjoint free random variable x ∈ (A, φ) is semicircular, if φ (xn ) = ωn cn , for all n ∈ N, (1.2.2) where ck =

1 k+1

2k k

=

1 k+1

(2k)! k!(2k − k)!

=

(2k)! k!(k + 1)!

are the k -th Catalan numbers for k ∈ N0 = N ∪ {0 }, and ωn = for all n ∈ N.

1 0

if n is even if n is odd,

Introduction ■ 5

obius inversion of Ref. [17], a self-adjoint free random By the M¨ variable x is semicircular in (A, φ), if and only if kn (x , ..., x ) = δn,2 ,

(1.2.3)

for all n ∈ N, where δ is the Kronecker delta. i.e., the free distributions of “all” semicircular elements are characterized by the free-moment sequence, (0, c1 , 0, c2 , 0, c3 , ...) , (1.2.4) equivalently, by the free-cumulant sequence, (0, 1, 0, 0, 0, ...),

(1.2.5)

by Eqs. (1.2.2) and (1.2.3), respectively. By the universality (1.2.4) and (1.2.5), the free distributions of all semicircular elements are called “the” semicircular law.

1.3

SOME FREE RANDOM VARIABLES INDUCED BY SEMICIRCULAR ELEMENTS

In this section, we consider two typical types of free random variables induced by semicircular elements. 1.3.1

Circular Elements

Let (B, ψ) be a topological ∗-probability space and assume that x1 and x2 are two, free, semicircular elements of (B, ψ). Then one can construct a new free random variable y ∈ (B , ψ), √ def 1 y = √ (xi + ix2 ) , with i = −1 . 2

(1.3.1)

Then, it has its adjoint, 1 y ∗ = √ (x1 − ix2 ) ∈ (B, ψ) , 2 since the semicircular elements x1 and x2 are self-adjoint in B . It shows that the free random variable y of Eq. (1.3.1) is not self-adjoint in (B, ψ). So, the free distribution of y is characterized by the joint free

6 ■ Constructive Analysis of Semicircular Elements

moments, or equivalently, by the joint free cumulants of {y, y ∗ }, by Eq. (1.1.1). Observe that, if y ∈ (B , ψ) is a free random variable (1.3.1), and if k• (...) is the free cumulant on B in terms of the linear functional ψ on B (e.g., [17,22,23]), then   kn y, y, ..., y  = kn

√1 2

(x1 + ix2 ) , ..., √12 (x1 + ix2 )

n-times

=

√1 2

n

kn ((x1 + ix2 ) , ..., (x1 + ix2 ))

by the bimodule-map property of free cumulant (e.g., [17,23]) =

√1 2

n

(kn (x1 , ..., x1 ) + kn (ix2 , ..., ix2 ))

since x1 and x2 are free, and hence, x1 and ix2 are free in (B, ψ) (Recall that if multi free random variables are free, then all “mixed” free cumulants of them vanish. e.g., [17,22]) =

√1 2

n

(δn,2 + i n kn (x2 , ..., x2 ))

by the semicircularity (1.2.3), or (1.2.5) of x1 =

√1 2

n

(δn,2 + in δn,2 ) =

by the semicircularity of x2   0  =  )  1( 2 =0 2 1+i

√1 2

n

δn,2 (1 + in )

if n = ̸ 2 if n = 2

= 0,

(1.3.2)

for all n ∈ N. Similar to Eq. (1.3.2), one can get that   kn y ∗ , y ∗ , ..., y ∗  = 0,

(1.3.3)

n-times

for all n ∈ N. The above free-cumulant computations (1.3.2) and (1.3.3) show that all n-th free cumulants of y and those of y ∗ vanish in (B, ψ). So,

Introduction ■ 7

to consider the free distribution of the free random variable y ∈ (B , ψ) of Eq. (1.3.1), it suffices to study “mixed” free cumulants of {y, y ∗ }, kn (y r1 , y r2 , ..., y rn ) , for all mixed n-tuples (r1 , ..., rn ) of {1, ∗}, for n ∈ N \ {1 }. Recall that (r1 , ..., rn ) is mixed in {1, ∗}, if and only if there exists at least one rs ∈ {r1 , ..., rn } such that rs ̸= rk in {1, ∗} for some k = ̸ s in {1, ..., n}. However, as it is discussed in Refs. [24,25], all mixed free cumulants of {y, y ∗ } vanish, except for the possible nonzero cases where kn (y, y ∗ , y, y ∗ , ..., y, y ∗ ) , and

(1.3.4) kn (y ∗ , y, y ∗ , y, ..., y ∗ , y) ,

by the R-diagonality of y (under the M¨ obius inversion of Ref. [17], and the semicircularity of x1 and x2 ) in (B, ψ). So, let’s concentrate on possible nonvanishing mixed free cumulants of {y, y ∗ } of Eq. (1.3.4). First, consider the case where n = 2 ; if n = 2 , then k2 (y, y ∗ ) = k2 √12 (x1 + ix2 ) , √12 (x1 − ix2 ) =

1 2

(k2 (x1 , x1 ) + k2 (ix2 , −ix2 ))

by the freeness of x1 and ±ix2 in (B, ψ) ( ) = 12 1 − i 2 k2 (x2 , x2 ) by the bi-module-map property of free cumulant =

1 2

(1 + 1 ) = 1 ,

(1.3.5)

and similarly, k2 (y ∗ , y) = 1 .

(1.3.6)

If n > 3 , then, by the semicircularity (1.2.3) and (1.2.5), kn (y, y ∗ , y, y ∗ , ..., y, y ∗ ) = 0, and

(1.3.7) kn (y ∗ , y, y ∗ , y, ..., y ∗ , y) = 0,

in (B, ψ).

8 ■ Constructive Analysis of Semicircular Elements

Therefore, the free distribution of a free random variable y ∈ (B , ψ) of Eq. (1.3.1) (or that of y ∗ ) is characterized by Eqs. (1.3.2), (1.3.3), (1.3.5), (1.3.6) and (1.3.7). Theorem 1.1: If y ∈ (B , ψ) is a free random variable (1.3.1), then the only nonvanishing joint free cumulants of {y, y ∗ } are k2 (y, y ∗ ) = 1 = k2 (y ∗ , y).

(1.3.8)

Proof . The proof of Eq. (1.3.8) is done by Eqs. (1.3.2), (1.3.3), (1.3.5), (1.3.6) and (1.3.7), with the help of Refs. [24,25]. The above theorem fully characterizes the free distribution of an element y ∈ (B , ψ) by the only nonvanishing joint free cumulants (1.3.8). Now, from Eq. (1.3.8), let’s study equivalent free-distributional data of a free random variable y ∈ (B , ψ) of Eq. (1.3.1). Observe that, for any n ∈ N, ψ (y n ) =

k|V | (y, ..., y) π∈N C(n)

V ∈π

by the M¨ obius inversion =

0 π∈NC (n)

= 0,

(1.3.9)

V ∈π

by Eq. (1.3.2), similarly, n

ψ ((y ∗ ) ) = 0,

(1.3.10)

for all n ∈ N. By Eqs. (1.3.8), (1.3.9) and (1.3.10), the only possible nonvanishing mixed free moments of {y, y ∗ } are n

n

ψ ((yy ∗ ) ) and ψ ((y ∗ y) ) , for all n ∈ N. Observe that n

ψ ((yy ∗ ) ) =

kV π∈NC (2n)

V ∈π

Introduction ■ 9

where kV are the block-depending free cumulants of {y, y ∗ }, by the M¨ obius inversion =

χB kB θ∈N C2 (2n)

B∈θ

by Eq. (1.3.8), where NC2 (2n) = {π ∈ NC (2n) : V ∈ π ⇔ |V | = 2 } , and χB =

      1

if a block B induces either kB =k2 (y, y ∗ ) or kB = k2 (y ∗ , y)

    0 

otherwise,

for all B ∈ θ, for all θ ∈ N C2 (2n), and hence, it goes to =

kB

=

B∈θ

θ∈χ2 (2n)

1 θ∈χ2 (2n)

= |χ2 (2n)| ,

(1.3.11)

B∈θ

by Eq. (1.3.8), where χ2 (2n) =

π ∈ N C2 (2n)

kV

∀V ∈ π induces either = k2 (y, y ∗ ), or kV = k2 (y ∗ , y)

.

Note that, since NC2 (2n) is defined over {1, ..., 2n}, such a subset χ2 (2n) of Eq. (1.3.11) is not empty for all n ∈ N. For example, χ2 (2) = {{(1, 2)}} = {12 } , χ2 (4) = {{(1, 2), (3, 4)}, {(1, 4), (2, 3)}} , and

(1.3.12)    {(1, 2), (3, 4), (5, 6)}, {(1, 6), (2, 3), (4, 5)},  {(1, 6), (2, 5), (3, 4)}, {(1, 2), (3, 6), (4, 5)}, χ2 (6) = ,   {(1, 4), (2, 3), (5, 6)}

etc., and hence,

10 ■ Constructive Analysis of Semicircular Elements

ψ (yy ∗ ) = 1, ψ (yy ∗ )

2

3

= 2, ψ (yy ∗ )

= 5,

respectively, by Eqs. (1.3.11) and (1.3.12). Theorem 1.2: Let y = √12 (x1 + ix2 ) ∈ (B , ψ) be a free random variable (1.3.1) generated by two, free, semicircular elements x1 , x2 ∈ (B , ψ). Then the only nonvanishing joint free moments are ψ ((yy ∗ )n ) = |χ2 (2n)| = ψ ((y ∗ y)n ) ,

(1.3.13)

for all n ∈ N, where χ2 (2n) is the subset (1.3.11) of the noncrossingpartition lattice NC (2n). Proof . It is not difficult to check that n

n

ψ ((y ∗ y) ) = |χ2 (2n)| = ψ ((yy ∗ ) ) , for all n ∈ N, similar to Eq. (1.3.11). Thus, the only nonvanishing freedistributional data (1.3.13) are obtained by Eqs. (1.3.9), (1.3.10) and (1.3.11). The above theorem re-characterizes the free distribution of a free random variable y ∈ (B , ψ) of Eq. (1.3.1), in terms of the joint free moments of {y, y ∗ }, which is equivalent to Eq. (1.3.8). Definition 1.2: A free random variable y = √12 (x1 + ix2 ) ∈ (B , ψ) of Eq. (1.3.1) is said to be the circular element generated by two, free, semicircular elements x1 and x2 Theorems 1.1 and 1.2 characterize the free distribution of a circular element y ∈ (B , ψ) by the only nonvanishing joint free cumulants, k2 (y, y ∗ ) = 1 = k2 (y ∗ , y) , by Eq. (1.3.8); and equivalently, by the only nonvanishing joint free moments, n

n

ψ ((yy ∗ ) ) = |χ2 (2n)| = ψ ((y ∗ y) ) , ∀n ∈ N, by Eq. (1.3.13).

Introduction ■ 11

By the universality (1.2.4) and (1.2.5) of “the” semicircular law, the free distributions of all circular elements are universal, too; we call them, “the” circular law, i.e., the circular law is characterized by Eq. (1.3.8) or Eq. (1.3.13). 1.3.2

Free Poisson Elements

Let (B, ψ) be a topological ∗-probability space, and let x ∈ (B , ψ) be a semicircular element. Assume that a self-adjoint free random variable a ∈ (B , ψ) is free from x in (B, ψ). i.e., all their “mixed” free cumulants of {a, x} vanish. By the self-adjointness of a ∈ (B , ψ), its free distribution is completely characterized by the free-moment sequence, ∞ (ψ (an ))n=1 , or, by the free-cumulant sequence,  

∞

kn a, a, ..., a n-times

,

n=1

by Eq. (1.2.1), where k• (...) is the free cumulant on B in terms of the linear functional ψ on B . Definition 1.3: Let x ∈ (B , ψ) be a semicircular element, and let a ∈ (B , ψ) be a self-adjoint free random variable, free from x . Define a free random variable Wxa ∈ (B , ψ) by def

Wxa = xax

in (B , ψ).

(1.3.14)

Then this free random variable Wxa ∈ (B , ψ) is called the free Poisson element generated by x and a. The free distribution of Wxa is called the free Poisson distribution of Wxa . By definition, if Wxa = xax ∈ (B , ψ) is a free Poisson element (1.3.14), then ∗ (Wxa ) = x∗ a∗ x∗ = xax = Wxa , in (B, ψ), and hence, it is a self-adjoint free random variable. So, the free Poisson distribution of it is fully characterized either by the free moments or by the free cumulants of Wxa by Eq. (1.1.2).

12 ■ Constructive Analysis of Semicircular Elements

Under the partial ordering (≤), the noncrossing-partition lattice NC (n) over {1, ..., n}, for n ∈ N, has its maximal element, 1n = {(1, 2, ..., n)} , the one-block partition, and its minimal element, 0n = {(1), (2), ..., (n)} , the n-block partition (e.g., [17,22,23]). Suppose {1, ..., n} = Ω1 ⊔ Ω2 , where Ωl are proper subsets of {1, ..., n}, for l = 1 , 2 , and ⊔ is the disjoint union, and let NC (Ωl ) be the noncrossing-patition lattice over Ωl , for l = 1 , 2 . Then, for θl ∈ NC (Ωl ), for l = 1 , 2 , one can construct a noncrossing partition, θ = θ1 ∨ θ2 ∈ N C(n), as the join of θ1 and θ2 , as in Refs. [17,22,23]. For example, if θ1 = {(2, 5), (3)} ∈ N C ({2, 3, 5}) , and θ2 = {(1, 6, 7) , (4)} ∈ N C ({1, 4, 6, 7}) , then θ1 ∨ θ2 = {(1, 6, 7), (2, 5), (3), (4)} ∈ N C(7). Now, fix n ∈ N, and let NC (3n) be the noncrossing-partition lattice over {1, ..., 3n}, and let Ω3n,1 = {1, 3, 4, 6, 7, 9, 10, ..., 3n − 3, 3n − 2, 3n} , and Ω3n,2 = {2, 5, 8, 11, ..., 3n − 1} , satisfying {1, ..., 3n} = Ω3n,1 ⊔ Ω3n,2 .

(1.3.15)

And then, take θ0 = {(1 , 3n), (3 , 4 ), (6 , 7 ), ..., (3n − 3 , 3n − 2 )} ∈ NC (Ω3n,1 ), (1.3.16) where Ω3n,1 is in the sense of Eq. (1.3.15).

Introduction ■ 13

Theorem 1.3: Let Wxa ∈ (B , ψ) be a free Poisson element (1.3.14). Then   kn Wxa , Wxa , ..., Wxa  = ψ (a n ) ,

(1.3.17)

n-times

for all n ∈ N. Proof . By the semicircularity (1.2.5) of x ∈ (B , ψ), kn (x, ..., x) = δn,2 , ∀n ∈ N. So, one has that kn (Wxa , ..., Wxa ) = kn (xax , ..., xax ) =

kπ (xax , ...xax ) π∈NC (Ω3n,2 ), π∨π0 ≤13n

where kπ (...) =

kV are the partition-depending free cumulants, and V ∈π

π0 = {(1, 2, 3), (4, 5, 6), ..., (3n − 2, 3n − 1, 3n)} ∈ N C(3n) (e.g., [17,22,23]), and hence, it goes to =

kπ∨θ0 (xax , ..., xax ) π∈NC (Ω3n,2 ), π∨θ0 ∈NC (3n)

where Ω3n,2 is in the sense of Eq. (1.3.15), and θ0 ∈ NC (Ω3n,1 ) is in the sense of Eq. (1.3.16)     =

kπ a, a, ..., a kθ0 x, x, x, ..., x π∈NC (Ω3n,2 ), π∨θ0 ∈NC (3n)

n-times

by the freeness of a and x (e.g., [17,22,23]) 

2n-times



n kπ a, a, ..., a (k2 (x , x ))

= π∈NC (Ω3n,2 ), π∨θ0 ∈NC (3n)

n-times

14 ■ Constructive Analysis of Semicircular Elements

=

kπ (a, ..., a) π∈NC (Ω3n,2 )

by the semicircularity of x =

kθ (a, ..., a) θ∈NC (n)

since two lattices NC (Ω3n,2 ) and NC (n) are isomorphic (or, equivalent) because |Ω3n,2 | = n = |{1, ..., n}| , and hence, it goes to = ψ (a n ), by the M¨ obius inversion. Therefore, the free-distributional data (1.3.17) holds. The above theorem characterizes the free Poisson distribution of a Wx ∈ (B , ψ) by Eq. (1.4.4). By the M¨ obius inversion of Ref. [17] and (1.4.4), one obtains the following equivalent result. Theorem 1.4: Let Wxa = xax ∈ (B , ψ) be a free Poisson element (12.1.1). Then n

ψ a|V |

ψ ((Wxa ) ) = π∈N C(n)

,

(1.3.18)

V ∈π

for all n ∈ N. Proof . For any n ∈ N, we have that  n

ψ ((Wxa ) ) =

  π∈NC (n)

V ∈π





  k|V | Wxa , Wxa , ..., Wxa  |V |-times

by the M¨ obius inversion of Ref. [17] ψ a |V |

= π∈NC (n)

by Eq. (1.3.17), for all n ∈ N.

V ∈π

,

Introduction ■ 15

Therefore, the free-moment computations (1.3.18) hold. The above two theorems fully characterize the free Poisson distribution of the free Poisson element Wxa ∈ (B , ψ) of Eq. (1.3.14). They illustrate that the free Poisson distribution of Wxa is determined by the free distribution of a in (B, ψ), either by (1.3.17) or by (1.3.18). They also illustrate that, different from the semicircularity and the circularity, free Poisson distributions are not universal. They depend on the free distributions of self-adjoint free random variables a ∈ (B , ψ), which are free from the given semicircular element x ∈ (B , ψ), inducing the free Poisson elements Wxa ∈ (B, ψ) of Eq. (1.3.14).

1.4

JOINT FREE MOMENTS OF MULTI SEMICIRCULAR ELEMENTS

In this section, we consider joint free distributions of mutually free, multi semicircular elements in a C ∗ -probability space (A, φ). If a given C ∗ -probabilistic structure (A, φ) is replaced to be a W ∗ -probability space, or a Banach ∗-probability space, the following results would be the same up to topology. Let X be an arbitrary finite set, and let N C (X) be the lattice of all noncrossing partitions over X, equipped with its partial ordering (≤) defined by def

π ≤ θ ⇐⇒ ∀V ∈ π, ∃U ∈ θ s.t., V ⊆ U, for “V 0X 1X

noncrossing partitions π, θ ∈ N C (X), where “V ∈ π” means that is a block of π.” Then this lattice NC (X ) has its minimal element whose blocks have a single element of X, and its maximal element which is the single-block partition. For instance, if Y = {a, b, c} ,

then N C (Y ) = {π1 , ..., π5 } , set-theoretically, where π1 = 0Y = {(a) , (b) , (c)} , π2 = {(a, b) , (c)} , π3 = {(a) , (b, c)} ,

16 ■ Constructive Analysis of Semicircular Elements

π4 = {(a, c) , (b)} , and π5 = 1Y = {(a, b, c)} , where the expressions (...) means the blocks of the partitions consisting of elements inside (...) . It is shown that 0Y = π1 ≤ πk ≤ π5 = 1Y , ∀k = 2, 3, 4. It is well known that |N C (X)| = c|X| , ∀ finite sets X (e.g., [22,23]), where |Z| is the cardinality of a set Z, and ck are the kth Catalan numbers for all k ∈ N0 = N∪{0}. So, in the above example, indeed, one can check that |N C (Y )| = c|Y | = c3 = 5. Now, let (B, ψ) be an arbitrary topological ∗-probability space, and y1 , ..., yn ∈ (B, ψ), free random variables, which are not necessarily mutually distinct from each other, for some n ∈ N. Suppose N C (n)

denote

=

N C ({1, ..., n})

is the noncrossing-partition lattice over {1, ..., n}, and π ∈ N C (n). Then, for a fixed partition π, one can define so-called the (noncrossing-) partition (free-)moment, ψπ (y1 , y2 , ..., yn ) of the free random variables y1 , ..., yn by ψπ (y1 , ..., yn ) =

ψV (y1 , ..., yn ) , V ∈π

where

(1.4.1) 

|V |

ψV (y1 , ..., yn ) = ψ 

 yil  ,

l=1

Introduction ■ 17

whenever ) ( V = i1 , ..., i|V | in π. For instance, if one has mutually distinct free random variables y1 , y2 , y3 ∈ (B, ψ) (which are not necessarily distinct from each other), and assume π1 = {(1, 2) , (3)} , π2 = {(1, 3) , (2)} ∈ N C (3) . Then ψπ1 (y1 , y2 , y3 ) = ψ (y1 y2 ) ψ (y3 ) , and ψπ2 (y1 , y2 , y3 ) = ψ (y1 y3 ) ψ (y2 ) , etc. Now, let x ∈ (B, ψ) be a semicircular element, i.e.,   ψ (xn ) = ωn c n2 , and knψ y, y, ..., y  = δn,2 , n-times

for all n ∈ N, where k•ψ (..) is the free cumulant on B with respect to the linear functional ψ, and δ is the Kronecker delta. Observe that    ωn c n2 = ψ (xn ) =

  π∈N C(n)

V ∈π

 ψ  k|V | x, x, x, ..., x |V |-times

by the M¨ obius inversion of Ref. [23] =

δ|V |,2 π∈N C(n)

V ∈π

=

1 U ∈θ

θ∈N C2 (m)

by the semicircularity, where N C2 (n) = {θ ∈ N C (n) : |U | = 2, ∀U ∈ θ} is the subset of N C (n) consisting of all “pair” noncrossing partitions whose blocks have only two elements, and hence, it goes to 1#(θ) =

= θ∈N C2 (m)

1 θ∈N C2 (m)

18 ■ Constructive Analysis of Semicircular Elements

where #(θ) is the number of blocks of a partition θ = |N C2 (n)| ,

(1.4.2)

for all n ∈ N. Lemma 1.1: Let x be a semicircular element of a topological ∗-probability space (B, ψ). Then ωn c n2 = ψ (xn ) = |N C2 (n)| = N C

n 2

,

(1.4.3)

for all n ∈ N, where ωk =

1 0

if k is even if k is odd,

for all k ∈ N. Proof . The formula (1.4.3) is shown by Eq. (1.4.2). The above formula (1.4.3) implies that, indeed, all “odd” free moments of a semicircular element vanish determined up to {ωk }k∈N of Eq. (1.4.3), since N C2 (n) = Ø = N C

n , whenever n is odd, 2

where Ø is the empty set. Corollary 1.1: Under the same hypothesis of Lemma 1.1,   ψ1n x, x, x, ..., x = ωn c n2 = |N C2 (n)| , ∀n ∈ N,

(1.4.4)

n-times

where 1n = 1{1,...,n} = {(1, ..., n)} is the maximal element of N C (n). Proof . Note that, by Eq. (1.4.1), one immediately obtain that   ψ1n x, x, ..., x = ψ (xn ) , ∀n ∈ N, n-times

Introduction ■ 19

because 1n = {((1, ..., n))} in N C (n), and hence, the formula (1.4.4) holds by Eq. (1.4.3). Now, let (A, φ) be a fixed C ∗ -probability space, and suppose there are N -many mutually free semicircular elements x1 , ..., xN ∈ (A, φ), for N ∈ N∞ >1 . By Ref. [23], all “mixed” free cumulants of x1 , ..., xN vanish by the freeness on them. And, by the self-adjointness of x1 , ..., xN in A, the free distribution ρ = ρx1 ,...,xN (1.4.5) of these semicircular elements is characterized by the joint free moments, ∞

∪

n=1

n

∪

(i1 ,...,in )∈{1,...,N }n

φ

xil

(1.4.6)

l=1

(e.g., see Refs. [17,22,23]). To verify the joint free distribution ρ of Eq. (1.4.5), we focus on computing free-distributional data of Eq. (1.4.6). For any s ∈ N, fix an s-tuple s

Is = (i1 , ..., is ) ∈ {1 , ..., N } .

(1.4.7)

For example, I8 = (1, 1, 3, 1, 3, 2, 2, 1) 8

is an 8-tuple of {1, 2, 3, 4, 5} . From the sequence Is of Eq. (1.4.7), define a set [Is ] by [Is ] = {i1 , ..., is } ,

(1.4.8)

without considering repetitions of quantities in {1, ..., N }. i.e., the set [Is ] induced by Is is a set with its cardinality s because we ignore the repetitions of quantities of {1, ..., N } in Is . For instance, if I8 is as above, then [I8 ] = {i1 , i2 , ..., i8 } , with its cardinality 8, satisfying i1 = i2 = i4 = i8 = 1 , and i3 = i5 = 3, and i6 = i7 = 2 ,

20 ■ Constructive Analysis of Semicircular Elements

without considering repetition; i.e., for instance, we regard all 1’s in I8 as mutually distinct elements i1 , i2 , i4 and i8 in the set [I8 ]. Then, from the set [Is ] of Eq. (3.8), one can define a “noncrossing” partition π(i1 ) of the noncrossing-partition lattice, NC ([Is ]) = NC ({i1 , i2 , ..., is }) (e.g., [17,22,23]), such that: (i) starting from i1 , construct the maximal block U1 of π(i1 ), satisfying U1 = ij1 = i1 , ij2 , ..., ijk

⇐⇒ ij1 = ij2 = ... = ijk = i1 ,

(1.4.9)

in {1, ..., N }, where k =| U1 | is the cardinality of the block U1 (as a set), (ii) construct the next noncrossing block U2 from the very next element, il ∈ [Is ] \ {ij1 , ..., ijk } = [Is ] \ U1 ; as in Eq. (1.4.9); do such processes until end, and (iii) the resulted partition π(i1 ) of (ii) is “maximal” for the partial ordering (≤) on the lattice NC ([Is ]) satisfying both (i) and (ii). For example, if I8 and [I8 ] are as above, then there exists a noncrossing partition, π(i1 ) = {(i1 , i2 , i4 , i8 ) , (i3 ) , (i5 ) , (i6 , i7 )} , in NC ([I8 ]), satisfying the above conditions (i), (ii) and (iii). Remark here that, even though i3 = i5 = 3 , one cannot have the block (i3 , i5 ) in π(i1 ) because this block (i3 , i5 ) has a crossing with the block U1 = (i1 , i2 , i4 , i8 ) of (1.4.9), so one has to take two separated blocks (i3 ) and (i5 ) to satisfy the conditions (i), (ii) and (iii). Now, let π(i1 ) ∈ N C ([Is ]) be the noncrossing partition induced by the first entry i1 of Is , and let π(i1 ) = {U1 , ..., Ut } , where t ≤ s, and Uk ∈ π(i1 ) are the blocks satisfying (i), (ii) and (iii), for k = 1 , ..., t. For example, if I8 is as above, then π(i1 ) = {U1 , U2 , U3 , U4 } with t = 4 < 8 = s, where U1 = {i1 , i2 , i4 , i8 } , U2 = {i3 } ,

Introduction ■ 21

and U3 = {i5 } , U4 = {i6 , i7 } . This partition π (i1 ) is regarded as the joint partition, π(i1 ) = 1{U1 } ∨ 1{U2 } ∨ ... ∨ 1{Ut } ,

(1.4.10)

where 1{Uk } in Eq. (1.4.10) are the maximal elements, the oneblock partitions, of NC (Uk ), for all k = 1 , ..., t, by regarding Uk as independent discrete sets. For example, if π(i1 ) for the first entry i1 of the above 8-tuple I8 is given as above, then π(i1 ) = 1{i1 ,i2 ,i4 ,i8 } ∨ 1{i3 } ∨ 1{i5 } ∨ 1{i6 ,i7 } . Now, construct a new noncrossing partition π(i2 ) ∈ NC ([Is ]) similarly for the second entry i2 of Is , as a noncrossing partition satisfying the conditions: (I) for the fixed entry i2 , construct the maximal block, B1 = ik1 , ..., ikm = i2 , ..., ikl , where l = |B1 | as in Eq. (1.4.9), satisfying ik1 = ... = ikm = ... = ikl = i2 , in {1, ..., N }, (II) do such processes for other entries step by step until end, and (III) the resulted partition π(i2 ) is maximal in N C ([Is ]) satisfying (I) and (II) altogether. For example, for the above 8-tuple I8 = (1, 1, 3, 1, 3, 2, 2, 1), we have π(i2 ) = {(i1 , i2 , i4 , i8 ) , (i3 ) , (i5 ) , (i6 , i7 )} . In this case, one can realize that this partition π(i2 ) is identically the same with π(i1 ) from the above example. Construct such noncrossing partitions for all rest entries i3 , ..., is of the s-tuple Is of Eq. (1.4.7). For example, from the above 8-tuple I8 , one has π(i1 ) = π(i2 ) = π(i4 ) = π(i8 ), as above, π(i3 ) = π(i5 ) = {(i3 , i5 ) , (i1 , i2 , i8 ) , (i4 ) , (i6 , i7 )} , and π(i6 ) = π(i7 ) = {(i6 , i7 ) , (i1 , i2 , i4 , i8 ) , (i3 ) , (i5 )} .

22 ■ Constructive Analysis of Semicircular Elements

Without loss of generality, one may write π(i1 ) = π(i2 ) = π(i4 ) = π(i8 ) π(i3 ) = π(i5 )

denote

=

denote

=

π(1),

π(3),

and

(1.4.11) π(i6 ) = π(i7 )

denote

=

π(2)!

Observation 1.1: It is not difficult to check that π(ij1 ) = π(ij2 ) in N C ([Is ]) ⇐⇒ ij1 = ij2 in {1, ..., N } , s

from the very constructions of our noncrossing partitions {π(il )}l=1 s induced by a fixed s-tuple Is = (il )l=1 of Eq. (3.7) in {1, ..., N }. s By the above observation, if Is = (i1 , ..., is ) ∈ {1, ..., N } is an s-tuple, and if it contains mutually distinct entries,

✥

ij1 < ij2 < ... < ijn in {1, ..., N } , where n ≤ N , then one can construct mutually distinct noncrossing partitions, π (ij1 ) , ..., π (ijn ) in N C ([Is ]) , (1.4.12) 8

satisfying the above conditions. For example, if I8 ∈ {1, ..., 5} is given as above, then one has mutually distinct partitions π(1), π(2) and π(3) in the sense of Eq. (1.4.11) (or, more generally, in the sense of Eq. (1.4.12)). s

Notation 1.1: (1) For a fixed s-tuple Is ∈ {1, ..., N } of Eq. (1.4.7), let π (ij1 ) , ..., π (ijn ) be the noncrossing partitions (1.4.12) in N C ([Is ]), for n ≤ N . Denote the subset of all such partitions by Π ([Is ]) = {π (ij1 ) , ..., π (ijn )} , in N C ([Is ]). (2) We denote the subset of Π ([Is ]) consisting of all partitions whose blocks have only even-many elements by Πe ([Is ]), i.e., Πe ([Is ]) = {θ ∈ Π ([Is ]) : |V | ∈ 2N, ∀V ∈ θ} ,

Introduction ■ 23

where 2N = {2n : n ∈ N}. ( ) (3) Let π ∈ N C ([Is ]) and suppose V = ik1 , ..., ik|V | ∈ π is a block. We say that “V has identical entries of Is ,” if ik1 = ik2 = ... = ik|V | in {1, ..., N } . i.e., a block V ∈ π has identical entries of Is , if and only if all elements of V are identically same in {1, ..., N }. For example, if I8 is given as above, then Π ([I8 ]) = {π(1), π(2), π(3)} is obtained as in Eq. (3.11). But,

✥

Πe ([Is ]) = Ø, because all distinct partitions π (1), π (2) and π (3) of Π ([I8 ]) contain blocks with odd-cardinalities; also, one can check that all blocks of the partitions in Π ([I8 ]) have identical entries of I8 . Let Is = (i1 , ..., is ) be an s-tuple (1.4.7), and let xi1 , ..., xis ∈ {x1 , ..., xN } be the corresponding semicircular elements of (A, φ) induced by Is . Define a free random variable X [Is ] by s def

xil ∈ (A, φ) .

X [Is ] =

(1.4.13)

l=1

For example, if I8 is as above, then X[I8 ] = x1 x1 x3 x1 x3 x2 x2 x1 = x21 x3 x1 x3 x22 x1 , which is a free reduced word with its length-6 in (A, φ) (e.g., [22,23]). If X [Is ] is a free random variable (1.4.13), then φ (X [Is ]) =

kV π∈N C([Is ])

V ∈π

by the M¨ obius inversion of Ref. [22], where kV = k|V | xik1 , ..., xik|V | , ( ) whenever V = ik1 , ..., ik|V | is a block of a partition π, where k• (...) is the free cumulant on A with respect to φ, and hence, =

kV θ∈P([Is ])

V ∈θ

24 ■ Constructive Analysis of Semicircular Elements

where

(1.4.14)

P ([Is ]) = {π ∈ N C ([Is ]) : V ∈ π has identical entries of Is , ∀V ∈ π} , in the sense of Notation 1.1, by the mutual-freeness of x1 , ..., xN in (A, φ) (i.e., all mixed free cumulants of x1 , ..., xN vanish by their mutual-freeness!) =

δn,2 θ∈P([Is ])

V ∈θ

by the semicircularity =

1 , θ∈P([Is ])∩NC2 ([Is ])

V ∈θ

by Eq. (1.4.14), where

(1.4.15)

def

NC2 (Y ) = {π ∈ NC (Y ) : V ∈ π ⇐⇒| V |= 2 } is the subset of the noncrossing-partition lattice N C(Y ) over finite sets Y . i.e., π ∈ N C2 (Y ), if and only if all blocks of π have two elements. By Eqs. (1.4.13) and (1.4.15), if there exists at least one k0 ∈ {1 , ..., t}, such that | Uk0 | is odd in N, then φ (X [Is ]) = 0.

(1.4.16)

So, the formula (1.4.15) is nonzero, only if (i) s ∈ N is even, and (ii) the s-tuple Is = (i1 , ..., is ) contains mutually distinct entries ik1 , ..., ikn (n ≤ N ), and each distinct entry appeared in Is even number of times. s

Lemma 1.2: Let Is ∈ {1, ..., N } be a fixed s-tuple (1.4.7) and let X[Is ] ∈ (A, φ) be the corresponding free random variable (1.4.13) N induced by the mutually free semicircular elements {xj }j=1 . Then φ (X[Is ]) = ωs |P2 ([Is ])| , where

(1.4.17) P2 ([Is ])

denote

=

P ([Is ]) ∩ N C2 ([Is ]) ,

and ωs and P ([Is ]) are in the sense of Eq. (1.4.14), respectively, for all s ∈ N.

Introduction ■ 25

Proof . Suppose s is odd in N. Then the formula (3.17) is obtained by Eq. (3.16). If s is even in N, then φ (X [Is ]) =

1 θ∈P([Is ])∩N C2 ([Is ])

V ∈θ

by Eq. (3.15) 1#(θ) =

= θ∈P([Is ])∩N C2 ([Is ])

1, θ∈P2 ([Is ])

where #(θ) is the number of blocks in θ ∈ N C ([Is ]). Therefore, the formula (1.4.17) holds true. Note that even though s ∈ N is even, it is possible that P2 ([Is ]) is empty. In such a case, φ (X [Is ]) = 0, anyway, by Eq. (1.4.17). Recall now the subset Π ([Is ]) of Notation 1.1, consisting of all mutually distinct partitions of Eq. (1.4.12). s

Lemma 1.3: Suppose Is = (i1 , ..., is ) ∈ {1, ..., N } be an s-tuple of Eq. (1.4.7) for s ∈ 2N, having its mutually distinct entries, ik1 < ik2 < ... < ikn with 1 ≤ n ≤ N, Let P2 ([Is ]) be in the sense of Eq. (1.4.17), and let Πe ([Is ]) be the subset of Notation 1.1 in N C ([Is ]). If P2 ([Is ]) ̸= Ø, and π ∈ P2 ([Is ]), then there exists θ ∈ Πe ([Is ]) such that π ≤ θ in the lattice N C ([Is ]). (1.4.18) If Πe ([Is ]) = ̸ Ø, then θ ∈ Πe ([Is ]) has at least one π ∈ P2 ([Is ]), such that π ≤ θ in N C ([Is ]), (1.4.19) if and only if all blocks of θ contain even-many elements. Proof . Under hypothesis, suppose P2 ([Is ]) = P ([Is ]) ∩ N C2 ([Is ])

26 ■ Constructive Analysis of Semicircular Elements

is not empty. Then, the subset Πe ([Is ]) of Π ([Is ]) is not empty by the very construction of Π ([Is ]) and its subset Πe ([Is ]). Moreover, for any π ∈ P2 ([Is ]), there exists θ ∈ Πe ([Is ]), such that π ≤ θ in N C ([Is ]), by the constructions of P2 ([Is ]). Therefore, the statement (1.4.18) holds. ̸ Ø, then P2 ([Is ]) = ̸ Ø because, for any Remark that if Πe ([Is ]) = θ ∈ Πe ([Is ]), one can take at least one π ∈ P2 ([Is ]) such that π ≤ θ in N C ([Is ]), by the definition of P ([Is ]) of Eq. (1.4.14). Thus, the statement (1.4.19) holds, too. The above lemma shows that if either P2 ([Is ]) or Πe ([Is ]) is not empty in N C ([Is ]), then all elements of P2 ([Is ]) are covered by elements of Πe ([Is ]) (in the sense of Eq. (1.4.18)), and conversely, all elements of Πe ([Is ]) cover elements of P2 ([Is ]). It means that if nonempty, Πe ([Is ]) classify P2 ([Is ]), i.e., if Πe ([Is ]) = {θ1 , ..., θn } , with 1 ≤ n ≤ N, then

n

P2 ([Is ]) = ⊔ P2 ([Is ] , θl ) , l=1

with

(1.4.20) P2 ([Is ] , θl ) = {π ∈ P2 ([Is ]) : π ≤ θl } ,

by Eqs. (1.4.18) and (1.4.19), for all l = 1, ..., n, where ⊔ is the disjoint union. Proposition 1.1: Let P2 ([Is ]) be in the sense of Eq. (1.4.14), and let s Πe ([Is ]) be in the sense of Notation 1.1, for Is ∈ {1, ..., N } , for s ∈ N. Then P2 ([Is ]) =

⊔ θ∈Πe ([Is ])

{π ∈ P2 ([Is ]) : π ≤ θ} .

(1.4.21)

Proof . The decomposition (1.4.21) of P2 ([Is ]) classified by the elements of Πe ([Is ]) is shown by Eq. (1.4.20). Using the above decomposition (1.4.21), the formula (1.4.17) is refined by the following result. Theorem 1.5: Let Is be an s-tuple (1.4.7), and X [Is ] ∈ (A, φ), the corresponding free random variable (1.4.13), and let Πe ([Is ]) be the

Introduction ■ 27

subset of N C ([Is ]) in the sense of Notation 1.1. Then φ (X[Is ]) =

c |V | θ∈Πe ([Is ])

V ∈θ

,

(1.4.22)

2

where Πe ([Is ]) is in the sense of Notation 1.1. Clearly, one has that φ (X [Is ]) = 0 if and only if Πe ([Is ]) = Ø. Proof . Under hypothesis, one has φ (X [Is ]) = ωs |P2 ([Is ])| by Eq. (1.4.17) = ωs

⊔

P2 ([Is ] , θ)

θ∈Πe ([Is ])

by Eq. (1.4.21), where P2 ([Is ] , θ) are in the sense of Eq. (1.4.20) |P2 ([Is ] , θ)|

= ωs θ∈Πe ([Is ])

= ωs

1 θ∈Πe ([Is ])

π∈P2 ([Is ]),π≤θ

1#(π)

= ωs θ∈Πe ([Is ])

π∈P2 ([Is ]),π≤θ

θ∈Πe ([Is ])

π∈P2 ([Is ]),π≤θ

V ∈π

θ∈Πe ([Is ])

π∈P2 ([Is ]),π≤θ

V ∈π

= ωs

1

= ωs

δ|V |,2

= ωs

φθ (xi1 , ..., xis ) ,

(1.4.23)

θ∈Πe ([Is ])

by the semicircularity, where φθ (•) are the partition moments of Eq. (1.4.1). And, by Eqs. (1.4.1), (1.4.3) and (1.4.4), one can get that φθ (xi1 , ..., xis ) =

φV (xi1 , ..., xis ) = V ∈θ

c |V | , V ∈θ

2

(1.4.24)

28 ■ Constructive Analysis of Semicircular Elements

for all θ ∈ Πe ([Is ]). Therefore, the free-distributional data (1.4.22) holds by Eqs. (1.4.23) and (1.4.24). So, by Eq. (1.4.22), if Πe ([Is ]) = Ø, then φ (X [Is ]) = 0. Conversely, if φ (X [Is ]) = 0, then Πe ([Is ]) is empty in Π ([Is ]) by Eqs. (1.4.23) and (1.4.24). Example: 1.1 (1) Let W = x21 x42 x1 x22 x1 ∈ (A, φ) be in the sense of Eq. (1.4.13). Then one can take the 10-tuple, let

IW = (1 , 1 , 2 , 2 , 2 , 2 , 1 , 3 , 3 , 1 ) = (i1 , ..., i10 ) , as in Eq. (1.4.7). Then the noncrossing partitions, π(1) = {(i1 , i2 , i7 , i10 ) , (i3 , i4 , i5 , i6 ) , (i8 , i9 )} , and π (2) = π (3) = π (1) in N C ([IW ]) , of Eq. (1.4.12) are obtained. So, we have Π ([IW ]) = {π (1)} = Πe ([IW ]) in the sense of Notation 1.1. Therefore, φ (W ) = c 42 c 24 c 22 = c22 c1 = 4, by Eq. (1.4.22). (2) Let Y = x12 x3 x1 x3 x22 x1 ∈ (A, φ). Then one can take an 8-tuple, let

IY = (1, 1, 3, 1, 3, 2, 2, 1) = (i1 , ..., i8 ) of Eq. (1.4.7). Thus, the corresponding partition, π(1) = {(i1 , i2 , i4 , i8 ) , (i3 ) , (i5 ) , (i6 , i7 )} , π (2) = π (1) , and π (3) = {(i1 , i2 ) , (i3 , i5 ) , (i4 ) , (i6 , i7 ) , (i8 )} ,

Introduction ■ 29

of Eq. (1.4.12) are obtained, and they construct Π ([IY ]) = {π (1) , π (3)} , however, Πe ([IY ]) = Ø in Π ([IY ]) , as in Notation 3.1 because all distinct partitions π(1) and π (3) contain odd blocks. Thus φ (Y ) = 0, by Eq. (1.4.22). (3) Let U = x1 x2 x21 x2 x12 x2 x12 x2 x1 ∈ (A, φ). Then one can take a 12-tuple, let

IU = (1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1) = (i1 , ..., i12 ) of Eq. (1.4.7). Thus, one obtains the noncrossing partitions, π (1) = {(i1 , i3 , i4 , i6 , i7 , i9 , i10 , i12 ) , (i2 ) , (i5 ) , (i8 ) , (i11 )} , and π (2) = {(i2 , i5 , i8 , i11 ) , (i1 , i12 ) , (i3 , i4 ) , (i6 , i7 ) , (i9 , i10 )} , constructing Π ([IU ]) = {π (1) , π (2)} , with Πe ([IU ]) = {π (2)} , since π (1) contains four odd blocks. Thus, we have φ (U ) = c 42 c 22 c 22 c 22 c 22 = c2 c14 = 2, by Eq. (1.4.22).

✥

30 ■ Constructive Analysis of Semicircular Elements

REFERENCES [1] M. Ahsanullah, Some Inferences on Semicircular Distribution, J. Stat. Theory Appl., 15, no. 3, (2016) 207–213. [2] H. Bercovici, and D. Voiculescu, Superconvergence to the Central Limit and Failure of the Cramer Theorem for Free Random Variables, Probab. Theory Related Fields, 103, no. 2, (1995) 215–222. [3] M. Bozejko, W. Ejsmont, and T. Hasebe, Noncommutative Probability of Type D, Int. J. Math., 28, no. 2, (2017) 30. [4] M. Bozheuiko, E. V. Litvinov, and I. V. Rodionova, An Extended Anyon Fock Space and Non-commutative Meixner-Type Orthogonal Polynomials in the Infinite-Dimensional Case, Uspekhi Math. Nauk., 70, no. 5, (2015) 75–120. [5] I. Cho, Semicircular Families in Free Product Banach *-Algebras Induced by p- Adic Number Fields over Primes p, Compl. Anal. Oper. Theory., 11, no. 3, (2017) 507–565. [6] I. Cho, Semicircular-Like Laws and the Semicircular Law Induced by Orthogonal Projections, Compl. Anal. Oper. Theory., 12 (2018) 1657– 1695. [7] I. Cho, Free Stochastic Integrals for Weighted-Semicircular Motion Induced by Orthogonal Projections. In: H. Dutta (ed.), Applied Mathematical Analysis: Theory, Methods, and Applications. Taylor & Francis: New York, 371–402 (2020). [8] I. Cho, Acting Semicircular Elements Induced by Orthogonal Projections on von Neumann Algebras, Mathematics, 5, (2017) 24. DOI: 10.3390/math5040074. [9] I. Cho, and J. Dong, Catalan Numbers and Free Distributions of Mutually Free Multi Semicircular Elements, Compl. Anal. Oper. Theory. (2020). Submitted to Compl. Anal. Oper. Theo. [10] I. Cho, and P. E. T. Jorgensen, Semicircular Elements Induced by Projections on Separable Hilbert Spaces. In: D. Alpay, and M. Vajiac (eds.), Operator Theory Advances & Applications. Birkhauser: Cham (2018). DOI: 10.1007/978-3-030-18484-1-6. [11] I. Cho, and P. E. T. Jorgensen, Banach *-Algebras Generated by Semicircular Elements Induced by Certain Orthogonal Projections, Opuscula Math., 38, no. 4, (2018) 501–535.

Introduction ■ 31 [12] I. Cho, and P. E. T. Jorgensen, Semicircular Elements Induced by p-Adic Number Fields, Opuscula Math., 35, no. 5, (2017) 665–703. [13] A. Connes, Noncommutative Geometry. Academic Press: San Diego, CA (1994). ISBN: 0-12-185860-X. [14] P. R. Halmos, Graduate Texts in Mathematics: Hilbert Space Problem Books. Springer (1982). ISBN: 978-0387906850. [15] I. Kaygorodov, and I. Shestakov, Free Generic Poisson Fields and Algebras, Comm. Alg., 46, no 4, (2018). DOI: 10.1080/00927872.2017. 1358269. [16] L. Makar-Limanov, and I. Shestakov, Polynomials and Poisson Dependence in Free Poisson Algebras and Free Poisson Fields, J. Alg., 349, no 1, (2012) 372–379. [17] A. Nica, and R. Speicher, Lectures on the Combinatorics of Free Probability, (1st Ed.), London Mathematical Society Lecture Note Series, 335. Cambridge University Press: Cambridge (2006). ISBN-13:9780521858526. [18] I. Nourdin, G. Peccati, and R. Speicher, Multi-Dimensional Semicircular Limits on the Free Wigner Chaos, Progr. Probab., 67, (2013) 211–221. [19] V. Pata, The Central Limit Theorem for Free Additive Convolution, J. Funct. Anal., 140, no. 2, (1996) 359–380. [20] F. Radulescu, Random Matrices, Amalgamated Free Products and Subfactors of the C*-Algebra of a Free Group of Nonsingular Index, Invent. Math., 115, (1994) 347–389. [21] F. Radulescu, Free Group Factors and Hecke Operators, notes taken by N. Ozawa, Proceeding of 24th Conference in Operator Theory, Theta Advanced Series in Mathematical, Theta Foundation, Univ. of the West Timisoara, Romania, 2012 (2014). [22] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Am. Math. Soc. Mem., 132, no. 627, (1998) x+88. [23] R. Speicher, A Conceptual Proof of a Basic Result in the Combinatorial Approach to Freeness, Infinit. Dimension. Anal. Quant. Prob. Related Topics, 3, (2000) 213–222. [24] R. Speicher, and T. Kemp, Strong Haagerup Inequalities for Free RDiagonal Elements, J. Funct. Anal., 251, no 1, (2007) 141–173.

32 ■ Constructive Analysis of Semicircular Elements [25] R. Speicher, and U. Haagerup, Brown’s Spectral Distribution Measure for R- Diagonal Elements in Finite Von Neumann Algebras, J. Funct. Anal., 176, no 2, (2000) 331–367. [26] V. S. Vladimirov, p-Adic Quantum Mechanics, Comm. Math. Phy., 123, no. 4, (1989) 659–676. [27] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet & East European Mathematics, vol. 1. World Scientific: Singapore (1994). ISBN: 978-981-02-0880-6. [28] D. Voiculescu, Aspects of Free Analysis, Jpn. J. Math., 3, no. 2, (2008) 163–183. [29] D. Voiculescu, Free Probability and the Von Neumann Algebras of Free Groups, Rep. Math. Phy., 55, no. 1, (2005) 127–133. [30] D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, vol. 1. American Mathematical Society: Ann Arbor, MI (1992). ISBN-13: 978-0821811405.

CHAPTER

2

Semicircular Elements Induced by p-Adic Number Fields Qp

n this section, we provide a motivation for the construction of semicircular elements in our future works (See Chapters 3 and 4). However, the main results of this section are not just providing a motivation; they, themselves, are independently interesting and applicable to many potential scientific areas, especially to non-Archimedean mathematical or physical research with statistical geometries of “very small distances.” For more about number-theoretic backgrounds, see Refs. [10,11,14]. Also, see Refs. [1–8] for fundamental applied connections among free probability, operator theory, operator algebra, and number theory. Throughout Chapter 2, we let Qp be the p-adic number fields for p ∈ P, where P is the set of all primes (or prime numbers) in the natural numbers (or positive integers) N. This Banach space Qp (in the set Q of all rational numbers) is also understood as a measure space, Qp = (Qp , σ (Qp ) , µp ) ,

I

equipped with the left-and-right additive invariant Haar measure µp on the σ-algebra σ (Qp ), consisting of all µp -measurable subsets. Recall also that Qp is a well-defined field algebraically. As a topological space, the p-adic number field Qp contains its basis elements,

DOI: 10.1201/9781003374817-2

33

34 ■ Constructive Analysis of Semicircular Elements

Uk = pk Zp = pk x : x ∈ Zp , ∀k ∈ Z, where Zp = x ∈ Qp : |x|p ≤ 1 ⊂ Qp is the unit disk, where |.|p is the p-norm on Q, satisfying the basis property, Qp = ∪ Uk , set-theoretically, k∈Z

and the chain property, · · · ⊂ U2 ⊂ U1 ⊂ U0 = Zp ⊂ U−1 ⊂ U−2 ⊂ · · · , and the measure-theoretic property, µp (Uk ) =

1 , pk

for all k ∈ Z. By regarding Qp as a measure space, one can define the commutative (pure-algebraic) ∗-algebra,   tS ∈ C, and   Mp = tS χS χS is a characteristic ,   function for S S∈σ(Qp ) equipped with an unbounded linear functional φp , f dµp , ∀f ∈ Mp .

φp (f ) = Qp

From this “unbounded”-measure-theoretic structure (Mp , φp ), we construct the corresponding C ∗ -algebra Mp (under a suitable representation of Mp ), and the bounded (or continuous) linear functionals {φp,j }j ∈Z on Mp , constructing “bounded” measure spaces, {(Mp , φp,j ) : j ∈ Z} . For fixed p ∈ P and j ∈ Z, the commutative Banach ∗-algebra LSp (j ), 0 and a linear functional τp,j are well-defined forming a measure space, (

) 0 LSp (j ), τp,j .

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 35

From the system, (

) 0 LSp (j) , τp,j : p ∈ P, j ∈ Z ,

of bounded-measure spaces, one can construct the “noncommutative” Banach ∗-algebra as the free-probabilistic free product ∗-algebra, LS =

⋆

p∈P, j∈Z

LSp (j) ,

over C, and the free product bounded linear functional acting on LS, τ0 =

0 τp,j .

⋆

p∈P, j∈Z

Then certain generating elemen ts of this noncommutative Banach ( ) 0 ∗-probability space LS, τ are weighted-semicircular (see below), and they naturally generate corresponding semicircular elements whose free distributions are the semicircular law. So, in this section, semicircular elements are adjointable Banachspace operators acting on a C ∗ -algebra induced by {Qp }p∈P , different from classical approaches.

2.1

ANALYSIS OF THE p-ADIC NUMBER FIELDS Qp

For more about p-adic number fields and p-adic analysis for p ∈ P, see Ref. [14] and cited papers therein. Throughout this section, we use the same notations and terminology with those of Ref. [14]. For a prime p ∈ P in N, let |.|p : Q → Q be the p-norm defined by 1 , pn

|x|p = |pn q|p =

(2.1.1)

whenever x = p n q ∈ Q, for some n ∈ Z, and q ∈ Q. For instance, 8 3 and

and

8 3 8 3

= 23 2

1 3

= 3−1 (8) 3

= p0 p

8 3

1 1 = , 23 8

= 2

3

=

1 3−1

= p

= 3,

1 = 1, p0

36 ■ Constructive Analysis of Semicircular Elements

for all p ∈ P \ {2, 3}. Then the p-norm |.|p is indeed a well-defined norm on Q, forming the corresponding normed space Q, |.|p . In this normed space, the maximal p-norm closure is the p-adic number field Qp . i.e., Qp is a Banach space under (2.1.1). Note that the p-norm is non-Archimedean satisfying |x1 + x2 |p = max |x1 |p , |x2 |p , for all x1 , x2 ∈ Q. Set-theoretically, one can understand Qp as a set, ∞

xl pl : xl = 0, 1, ..., p − 1, ∀l .

Qp = ∪

N ∈N

(2.1.2)

l=−N

Then, under the p-adic addition (+), and the p-adic multiplication (·) of Ref. [14] (which are defined like the polynomial addition, and the polynomial multiplication in p), this set Qp of Eq. (2.1.2) forms a field, algebraically. i.e., the p-adic number field Qp is a Banach field. Now, let’s understand the Banach space Qp , as a measure space, Qp = (Qp , σ (Qp ) , µp ) , where σ (Qp ) is the σ-algebra of Qp , consisting of all µp -measurable subsets, where µp is the left-and-right additive invariant Haar measure on Qp , satisfying µp (Zp ) = 1, where

(2.1.3) Zp = x ∈ Qp : |x|p ≤ 1

is the unit disk of Qp . Define the subsets {Uk }k∈Z of Qp by Uk = p k Zp = p k x ∈ Qp : x ∈ Zp , for all k ∈ Z.

(2.1.4)

Then, by Eqs. (2.1.1) and (2.1.3), they are measurable subsets satisfying 1 µp (Uk ) = k , for all k ∈ Z, p

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 37

and hence,

(2.1.5) µp (x + Uk ) =

1 = µp (Uk + x) , ∀x ∈ Qp , pk

for all k ∈ Z, by the invariance of µp . Then, by Eqs. (2.1.2) and (2.1.4), as a topological space, Qp satisfies Qp = ∪ Uk , k∈Z

and

(2.1.6) · · · ⊂ U2 ⊂ U1 ⊂ U0 = Zp ⊂ U−1 ⊂ U−2 ⊂ · · · .

Now, motivated by Eq. (2.1.6), define new measurable subsets {∂k }k∈Z of Qp by ∂k = Uk \ Uk+1 , for all k ∈ Z. (2.1.7) Then, by Eqs. (2.1.5) and (2.1.7), µp (∂k ) = µp (Uk ) − µp (Uk+1 ) =

1 1 − k+1 , k p p

and

(2.1.8) Qp = ⊔ ∂k , k∈Z

by Eq. (2.1.6), where ⊔ is the disjoint union, since  if k1 = k2 in Z  ∂k1 ∂k1 ∩ ∂k2 =  Ø if k1 ̸= k2 in Z, where o ¨ means the empty set. The measurable subsets ∂k are called the k -th boundaries of Qp , for all k ∈ Z. Define a set Mp by     Mp = tS χS : tS ∈ C , (2.1.9)   S∈σ(Qp )

where χS are the characteristic functions of the measurable subsets S ∈ σ (Qp ), and is the finite sum. So, algebraically, the set Mp

38 ■ Constructive Analysis of Semicircular Elements

forms an algebra of measurable functions induced by σ (Qp ), over the complex numbers C. One can naturally define a unary operation (∗) on the algebra Mp of Eq. (2.1.9) by  ∗ tS χS  =

 S∈σ(Qp )

tS χS ,

(2.1.10)

S∈σ(Qp )

where z are the conjugates of z ∈ C. It is not difficult to check that the operation (2.1.10) is a welldefined adjoint on Mp , since f ∗∗ = f,

∀f ∈ Mp ,

and ∗

(zf ) = zf ∗ , ∀z ∈ C, and f ∈ Mp , and ∗

(f1 + f2 ) = f1∗ + f2∗ , ∀f1 , f2 ∈ Mp , and ∗

(f1 f2 ) = f2∗ f1∗ , ∀f1 , f2 ∈ Mp . i.e., the algebra Mp of Eq. (2.1.9) is a ∗-algebra equipped with its adjoint (2.1.10) over C. On this ∗-algebra Mp , define a linear functional, φp : Mp → C, by

(2.1.11) f dµp ∀f ∈ Mp ,

φp (f ) = Qp

i.e., 



φp 

tS χS  =

S∈σ(Qp )

tS µp (S) . S∈σ(Qp )

Observe that  φp 

 tS χS  =

S∈σ(Qp )

tS S∈σ(Qp )

χS dµp Qp

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 39

=

tS

χS d µp ⊔ ∂k

S ∈σ (Qp )

k ∈Z

by Eq. (2.1.8) =

tS

χS dµp k∈Z

S∈σ(Qp )

=

µp (S ∩ ∂k )

tS S∈σ(Qp )

=

k∈Z

µp (S ∩ ∂k ) ,

tS

S∈σ(Qp )

∂k

k∈ΛS

where

(2.1.12) ΛS = {k ∈ Z : S ∩ ∂k ̸= Ø} ,

for all S ∈ σ (Qp ). tS χS ∈ Mp be a measurable function.

Proposition 2.1: Let S ∈σ(Qp )

Then  φp 

 tS χS  =

S∈σ(Qp )

µp (S ∩ ∂k ) ,

tS S∈σ(Qp )

k∈ΛS

where

(2.1.13) ΛS = {k ∈ Z : S ∩ ∂k = ̸ Ø} , ∀S ∈ σ (Qp ) .

Proof . The proof of Eq. (2.1.13) is done by Eq. (2.1.12). By the above proposition, the following corollary is immediately obtained. Corollary 2.1: Let S ∈ σ (Qp ). Then there exist the quantities, {rj ∈ R : 0 ≤ rj ≤ 1, ∀j ∈ ΛS } ,

40 ■ Constructive Analysis of Semicircular Elements

such that

(2.1.14) φp (χS ) =

1 1 − j+1 pj p

rj j∈ΛS

,

where ΛS ⊂ Z is in the sense of Eq. (2.1.13). Proof . By Eq. (2.1.13), µp (S ∩ ∂j ) ,

φp (χS ) = j∈ΛS

where 0 ≤ µp (S ∩ ∂j ) ≤ µp (∂j ) =

1 1 − j+1 , j p p

for all j ∈ ΛS . Thus, the computation (2.1.14) holds. By Eq. (2.1.14), one obtains the following special-but-important computation. Corollary 2.2: For all j ∈ Z, we have ( ) 1 1 φp χ∂j = j − j+1 . p p Proof . It is trivial that Λ∂j = {j} in Z formula (2.1.15) holds by Eq. (2.1.14). Also, the formula (2.1.14) makes functional φp is not bounded on Mp . contains its unity (or the multiplication

(2.1.15)

by Eq. (2.1.8). Therefore, the us confirm that the linear Note that the ∗-algebra Mp identity),

1Mp = χQp , and it satisfies that ( ) φp χQp = k∈Z

1 1 − k+1 pk p

= lim (pn ) → ∞, n→∞

by Eqs. (2.1.13) and (2.1.5). So, one can have an unbounded-measuretheoretic model, (Mp , φp ), for all p ∈ P.

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 41

2.2

STATISTICAL MODELS INDUCED BY Qp

Throughout this section, let’s fix a prime p ∈ P, and Qp , the padic number field, and let (Mp , φp ) be the corresponding unboundedmeasure-theoretic model.  Let Uk = p k Zp k ∈Z be the basis elements (2.1.4), and let {χUk }k∈Z be the corresponding measurable functions of Mp , satisfying φp (χUk ) = µp (Uk ) =

1 , pk

by Eq. (2.1.5), or by Eqs. (2.1.13), or (2.1.14) for all k ∈ Z. Let χUk1 , χUk2 ∈ Mp be the characteristic functions of basis elements Uk1 and Uk2 , respectively. Then χUk1 χUk2 = χUk1 ∩Uk2 = χUmax {k1 ,k2 } ,

(2.2.1)

by the chain property (2.1.6), where  max {k1 , k2 } =

k1 k2

if k1 ≥ k2 if k1 < k2 ,

for all k1 , k2 ∈ Z. Proposition 2.2: Let Ukl = pkl Zp be basis elements (2.1.4) of Qp , and χUkl ∈ Mp , the characteristic functions of them, for l = 1 , ..., n, for n ∈ N. Then ! n Y 1 (2.2.2) φp χUkl = max{k ,...,k } . 1 n p l=1 Proof . By the induction on Eq. (2.2.1), one has n Y

χUkl = χUrmmax{k1 ,...,kn } in Mp .

l=1

P

So, by Eqs. (2.1.5), (2.1.13) and (2.1.14), the formula (2.2.2) holds. Now, let {∂k }k∈Z be the boundaries of Qp , and let {χ∂k }k∈Z ⊂ Mp be the corresponding characteristic functions. Then χ∂k1 χ∂k2 = χ∂k1 ∩∂k2 = δk1 ,k2 χ∂k1 in Mp ,

(2.2.3)

42 ■ Constructive Analysis of Semicircular Elements

where δ is the Kronecker delta, since ∂k1 ∩ ∂k2 =

∂k1 = ∂k2 Ø

if k1 = k2 otherwise,

by Eq. (2.1.8), for all k1 , k2 ∈ Z. Proposition 2.3: Let ∂kl be the kl -th boundaries of Qp , and χ∂kl ∈ Mp , the corresponding characteristic functions, for l = 1 , ..., n, for n ∈ N. Then  1 − pk11+1 if k1 = k2 = ... = kn  n pk1 φp χ∂kl = (2.2.4)  l=1 0 otherwise. Proof . By the induction on Eq. (2.2.3), n

( ) χ∂kl = δk1 ,k2 δk2 ,k3 · · · δkn−1 ,kn χ∂k1 , l=1

in Mp . Therefore, the formula (2.2.4) holds by Eq. (2.1.13) or (2.1.14). Also, by Eq. (2.2.3), one can realize that the measurable functions {χ∂k }k∈Z forms a family of mutually orthogonal projections of Mp in the sense that: ∗

2

(χ∂k ) = χ∂k = χ∂k ∩∂k = χ∂k χ∂k = (χ∂k ) , and

(2.2.5) χ∂k1 χ∂k2 = δk1 ,k2 χ∂k1 ,

in Mp , for all k , k1 , k2 ∈ Z. Observe now that, if S1 , S2 ∈ σ (Qp ), then χS1 χS2 = χS1 ∩S2 =

χ(S1 ∩S2 )∩∂j j∈ΛS1 ∩S2

=

χ(S1 ∩S2 )∩∂j = j∈ΛS1 ∩S2

=

χ(S1 ∩∂j )∩(S2 ∩∂j ) j∈ΛS1 ∩S2

χ(S1 ∩∂j ) χ(S2 ∩∂j ) j∈ΛS1 ∩S2

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 43

= j∈ΛS1 ∩ΛS2

=

χ(S1 ∩∂j ) χ(S2 ∩∂j )

χS1 ∩∂j j∈ΛS1

χS2 ∩∂j ,

(2.2.6)

j∈ΛS2

where ΛS ⊆ Z are in the sense of Eq. (2.1.13), for all S ∈ σ (Qp ). Proposition 2.4: Let Sl ∈ σ (Qp ) be measurable sets, and χSl ∈ Mp , the corresponding characteristic functions, for l = 1 , ..., n, for n ∈ N. Then there exists a subset n

ΛS1 ,...,Sn = ∩ ΛSl ⊆ Z, l=1

where ΛSl are in the sense of Eq. (2.1.13), and the quantities 0 ≤ rj ≤ 1 , ∀j ∈ ΛS1 ,...,Sn , such that n

φp

χSl

=

1 1 − j+1 pj p

rj j∈ΛS1 ,...,Sn

l=1

.

(2.2.7)

Proof . By the induction on Eq. (2.2.6), one has n

n

χSl = l=1

χSl ∩∂j j∈ΛS1 ,...,Sn

,

l=1

and hence, n

χSl = l=1

χ j∈ΛS1 ,...,Sn

n

∩ Sl ∩∂j

,

l=1

in Mp . Therefore, by Eq. (2.1.13), there exist   0 ≤ rj ≤ 1, for all   rj ,   j ∈ ΛS1 ,...,Sn satisfying the formula (2.2.7). Of course, if the subset ΛS1 ,...,Sn is empty, then the above formula (2.2.7) vanish.

44 ■ Constructive Analysis of Semicircular Elements

2.3

REPRESENTATIONS OF (Mp , φp )

In this section, we fix a prime p ∈ P, and the corresponding unbounded-measure space (Mp , φp ). By understanding the p-adic number field Qp as a measure space, construct the L2 -Hilbert space, def

Hp = L2p (Qp , σ (Qp ) , µp ) = L2 (Qp ) .

(2.3.1)

i.e., the Hilbert space Hp consists of all square-integrable functions h on Qp , 2

|h| dµp < ∞, Qp

under the inner product, h1 h2∗ dµp , ∀h1 , h2 ∈ Hp ,

⟨h1 , h2 ⟩2 = Qp

inducing the L2 -norm, ∥h∥2 =

⟨h, h⟩2 , ∀h ∈ Hp .

Then, by the definition (2.1.9), the ∗-algebra Mp canonically act on the Hilbert space Hp of Eq. (2.3.1), via an action, αp : Mp →B (Hp ) , where B (Hp ) is the operator algebra consisting of all bounded (linear) operators on Hp , i.e., for all f ∈ Mp , (αp (f )) (h) = f h, ∀h ∈ Hp .

(2.3.2)

Then, indeed, the action αp of Eq. (2.3.2) is a well-defined ∗-algebraaction of Mp acting on the Hilbert space Hp of Eq. (2.3.1), since ∥αp (f ) (h)∥2 = ∥f h∥2 ≤ ∥f ∥2 ∥h∥2 < ∞, for all f ∈ Mp and h ∈ Hp because of the (pure-algebraic) definition of Mp (under finite-sum) implying that ∥f ∥2 < ∞, ∀f ∈ Mp ;

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 45

and (αp (f1 f2 )) (h) = f1 f2 h = f1 (f2 h) = f1 (αp (f2 )(h)) = (αp (f1 )αp (f2 )) (h), for all f1 , f2 ∈ Mp , for all h ∈ Hp , implying that αp (f1 f2 ) = αp (f1 ) αp (f2 ) , ∀f1 , f2 ∈ Mp ,

(2.3.3)

on Hp , by Eq. (2.3.2); and ⟨αp (f ∗ ) (h) , h⟩2 = ⟨f ∗ h, h⟩2 =

f ∗ hh∗ dµp Qp ∗

hh∗ f ∗ dµp =

= Qp

h (f h) dµp Qp

= ⟨h, fh⟩2 = ⟨h, αp (f ) (h)⟩2 , implying that ∗

∗

αp (f ∗ ) = αp (f ) ⇐⇒ (αp (f )) = αp (f ∗ ) ,

(2.3.4)

on Hp , for all f ∈ Mp . Proposition 2.5: Let Hp = L2 (Qp ) be the Hilbert space (2.3.1), and let αp be a morphism (2.3.2). Then the pair (Hp , αp ) forms a welldefined Hilbert-space representation of the ∗-algebra Mp . In other words, Mp acts on Hp (or, in B (Hp )) via αp . Proof . It is enough to show that the morphism αp of Eq. (2.3.2) is a well-defined ∗-homomorphism from Mp into the operator algebra B (Hp ). However, by Eqs. (2.3.3) and (2.3.4), it is indeed. Therefore, the pair (Hp , αp ) is a Hilbert-space representation of Mp . The above proposition shows that our statistical model (Mp , φp ) acts naturally on the Hilbert space Hp of Eq. (2.3.1). Definition 2.1: Let (Hp , αp ) be the Hilbert-space representation of the L2 -Hilbert space Hp of Eq. (2.3.1) and the ∗-algebra-action αp of Eq. (2.3.2). Then it is called the p-Hilbert-space representation of Mp . In particular, the Hilbert space Hp is called the p-Hilbert space for p ∈ P.

46 ■ Constructive Analysis of Semicircular Elements

Depending on the p-Hilbert-space representation (Hp , αp ) of Mp , define the C ∗ -subalgebra Mp of the operator algebra B (Hp ) by def

Mp = C ∗ (αp (Mp )) = C [αp (Mp )],

(2.3.5)

where C ∗ (Y ) mean the C ∗ -subalgebras generated by subsets Y of B (Hp ), and C [Z] mean the (pure-algebraic) algebras generated by the subsets Z of B (Hp ), and where Y are the C ∗ -topological closures of subsets Y . Definition 2.2: Let Mp be the C ∗ -subalgebra (2.3.5) of the operator algebra B (Hp ), induced by the ∗-algebra Mp . Then it is called the p-C ∗ -algebra of Mp (in terms of (Hp , αp )). Notation 2.1: From below, for convenience, we denote αp (f ) ∈ Mp by f p . Also, we denote αp (χS ) by αSp , for all S ∈ σ (Qp ); and, especially, αp (χ∂k ) = α∂pk

denote

=

αkp ,

for all boundaries {∂k }k∈Z ⊂ σ (Qp ). By the definition (2.3.5), all elements T of Mp are expressed by X T = tS αSp , in Mp , with tS ∈ C, S∈σ(Qp )

where is an infinite (or a C ∗ -limits of finite) sum(s), where αSp are in the sense of Notation 2.1. P

2.4

MEASURE-THEORETIC MODELS ON {Mp }p∈P

In this section, we fix a prime p ∈ P. Similar to the unboundedmeasure-theoretic model (Mp , φp ) of Sections 2.2 and 2.3, we construct-and-study measure-theoretic models, {(Mp , φp,j ) : j ∈ Z} , for the p-C ∗ -algebra Mp of Eq. (2.3.5). Especially, they would be “bounded-”measure-theoretic models. Define linear functionals φp,j : Mp → C by X 

 def

φp,j (T ) = T χ∂j , χ∂j 2 = tS χS∩∂j , χ∂j 2 , (2.4.1) S∈σ(Qp )

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 47

for all tS αSp ∈ Mp ,

T =

S∈σ(Qp )

for all j ∈ Z (under Notation 2.1), where ⟨, ⟩2 is the inner product on the p-Hilbert space Hp of Eq. (2.3.1). Observe that, for any T = tS αSp ∈ Mp , S ∈σ(Qp )

φp,j (T ) =

χS χ∂j χ∗∂j dµp

tS Qp

S∈σ(Qp )

by Eqs. (2.3.1) and (2.4.1) =

tS S∈σ(Qp )

χS χ∂j dµp Qp

tS µp (S ∩ ∂j ) ,

=

(2.4.2)

S∈σ(Qp )

implying that

|φp,j (T )| ≤

tS S∈σ(Qp )

1 1 − j+1 pj p

by Eqs. (2.1.14) and (2.4.2), if and only if  1 1 |φp,j (T )| ≤ − j+1  pj p

,

 tS  ,

S∈σ(Qp )

if and only if

(2.4.3) |φp,j (T )| ≤

1 1 − j+1 j p p

∥T ∥ ,

by Eqs. (2.4.2), where ∥.∥ means the C ∗ -norm on Mp (inherited from the operator norm on B (Hp )). Since T ∈ Mp is arbitrary, the inequality (2.4.3) shows that the linear functional φp,j is bounded on Mp . Also, one can consider the

48 ■ Constructive Analysis of Semicircular Elements p boundedness of φp,j of Eq. (2.4.1) by considering the unity 1Mp = αQ p of Mp . Indeed,

( ) 1 1 φp,j 1Mp = µp (∂j ) = j − j+1 < ∞, p p

(2.4.4)

by Eq. (2.1.15). Thus, one can re-characterize the boundedness (2.4.3) of the linear functional φp,j , by Eq. (2.4.4). The mathematical pairs {(Mp , φp,j ) : j ∈ Z} form the (commutative) measure-theoretic structures for a fixed prime p ∈ P. Definition 2.3: Let p ∈ P be a fixed prime, and Mp , the corresponding p-C ∗ -algebra. The bounded-measure-theoretic structures, (Mp , φp,j ) , for j ∈ Z, are called the j -th (boundary-)p-(C ∗ -)measure spaces. Now, fix j ∈ Z, and the corresponding j -th p-measure space (Mp , φp,j ). Proposition 2.6: Let T = S ∈F

tS αSp ∈ (Mp , φp,j ), where F ⊆ σ (Qp ).

Then there exists rT ∈ C, such that rT =

tS in C, S∈T

and

(2.4.5) φp,j (T ) = rT

1 1 − j+1 pj p

,

where T = ∪ ΛS∩∂j in Z, S∈F

where ΛY are in the sense of Eq. (2.1.13) for all Y ∈ σ (Qp ). Proof . The statistical data (2.4.5) is shown by Eqs. (2.2.7) and (2.4.2). By Eq. (2.4.5), the following corollary is obtained.

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 49

Corollary 2.3: For any S ∈ σ (Qp ), there exists 0 ≤ rS ≤ 1 in R, such that 1 1 φp,j (αSp ) = rS − j+1 . (2.4.6) j p p

Proof . The formula (2.4.6) is immediately obtained by Eq. (2.4.5). By Eq. (2.4.6), we have the following special case. Corollary 2.4: For any k ∈Z, if αkp = α∂pk = αp (χ∂k ) ∈ Mp , then φp,j (αkp ) = δj,k

1 1 − j+1 j p p

.

(2.4.7)

Proof . The formula (2.4.7) is proven by Eq. (2.4.6), since  if j = k  ∂j Λ∂k ∩∂j =  Ø otherwise, for all k ∈ Z.

2.5

C ∗ -SUBALGEBRAS Sp OF (Mp , φp,j )

Throughout this section, we fix a prime p ∈ P, and the corresponding p-C ∗ -algebra Mp , inducing j -th p-measure spaces, {(Mp , φp,j ) : j ∈ Z} . Here, let’s focus on the elements, {αkp ∈ Mp : k ∈ Z} . One can observe that ( ) ∗ ∗ (αkp ) = αp (χ∂k ) = αp χ∗∂k = αp (χ∂k ) = αkp , and

(2.5.1) 2

(αkp ) = αp (χ∂k χ∂k ) = αp (χ∂k ∩∂k ) = αkp ,

50 ■ Constructive Analysis of Semicircular Elements

for all k ∈ Z. Moreover, for any k1 , k2 ∈ Z, we have ( ) αkp1 αkp2 = αp χ∂k1 ∩∂k2 = δk1 ,k2 αkp1 ,

(2.5.2)

in Mp . i.e., all element αkp induced by the k -th boundaries ∂k of Qp are mutually orthogonal projections of Mp , by Eqs. (2.5.1) and (2.5.2). Definition 2.4: Define the C ∗ -subalgebra Sp of the p-C ∗ -algebra Mp by Sp = C ∗ ({αkp : k ∈ Z}) = C ({αkp : k ∈ Z}), (2.5.3) over C = C · 1Mp , where Z are the C ∗ -norm closures of the subsets Z of Mp . This C ∗ -subalgebra Sp is called the p-boundary (C ∗ -)subalgebra of Mp . By the definition (2.5.3), the p-boundary subalgebra Sp is characterized as follows. Proposition 2.7: Let Sp be the p-boundary subalgebra of the p-C ∗ algebra Mp . Then ∗-iso

Sp =

∗-iso

⊕ (C · αkp ) = C⊕|Z| ,

(2.5.4)

k ∈Z

in Mp , where ⊕ is the direct product of C ∗ -algebras. Proof . Let Sp be the p-boundary subalgebra of Mp . Since it is generated by the family {αkp ∈ Mp : k ∈ Z} of mutually orthogonal projections by Eqs. (2.5.1) and (2.5.2), the isomorphism theorem (2.5.4) holds in Mp . Proposition 2.7 characterizes the C ∗ -algebra Sp by Eq. (2.5.4). It shows that the p-boundary subalgebra Sp acts like a diagonal subalgebra of Mp . So, by definition, every element S of Sp has its expression, S= tk αkp , with tk ∈ C. k∈Z

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 51

Also, the above proposition illustrates that, for this C ∗ -subalgebra Sp , one can have the bounded-measure-theoretic sub-structures, (

) Sp , φp,j = φp,j |Sp : j ∈ Z ,

(2.5.5)

from the bounded-measure-theoretic structures, {(Mp , φp,j ) : j ∈ Z} , the j -the p-measure spaces. Proposition 2.8: Let S = k ∈Z

tk αkp ∈ (Sp , φp,j ), where (Sp , φp,j ) is in

the sense of Eq. (2.5.5) for an arbitrarily fixed j ∈ Z. Then φp,j (S) = tj

1 1 − j+1 j p p

.

(2.5.6)

Proof . By Eqs. (2.4.5), (2.4.6), (2.4.7) and (2.5.4), if S ∈ Sp is given as above, then φp,j (S) =

tk δj,k k∈Z

1 1 − j+1 j p p

,

implying the formula (2.5.6). By Eq. (2.5.6), it is obtained that φp,j (αkp ) = δj,k

1 1 − j+1 j p p

, ∀k ∈ Z,

(2.5.7)

for all generators {αkp }k∈Z of Sp , for j ∈ Z.

2.6

STATISTICAL MODELS FOR BANACH ∗-ALGEBRAS LSp (j)

Let p ∈ P be an arbitrarily fixed prime, and Sp , the p-boundary subalgebra (2.5.3) of the p-C ∗ -algebra Mp , satisfying the structure theorem (2.5.4), which induces the bounded-measure-theoretic structures, {(Sp , φp,j ) : j ∈ Z} ,

52 ■ Constructive Analysis of Semicircular Elements

of Eq. (2.5.5). By Eqs. (2.5.6) and (2.5.7), the generating projections {αkp }k∈Z of Sp satisfy φp,j (αkp ) = δj,k

1 1 − j+1 j p p

=

δj,k pj

1−

1 p

,

(2.6.1)

for all k ∈ Z, for any fixed j ∈ Z. Recall the Euler totient function, an arithmetic function, ϕ : N → C, defined by def

ϕ (n) = |{m ∈ N : gcd (n, m) = 1}| ,

(2.6.2)

where |Y | mean the cardinalities of sets Y , and gcd (, ) is the greatest common divisor. It is well known that, for any n ∈ N, 1−

ϕ (n) = n q∈P, q|n

1 q

,

where “q | n” means “q divides n.” For instance, ϕ(q) = q − 1 = q 1 −

1 q

, ∀q ∈ P.

(2.6.3)

by Eq. (2.6.2). Thus, by Eqs. (2.6.1) and (2.6.3), φp,j (αkp ) =

δj ,k p j +1

p 1−

1 p

= δj ,k

ϕ (p) , p j +1

(2.6.4)

for all k ∈ Z, where ϕ is the Euler totient function (2.6.2), for j ∈ Z. Motivated by Eq. (2.6.4), define new linear functionals ψp,j : Sp → C by the linear morphism on Sp , satisfying def

ψp,j =

1 ϕ (p)

φp,j , on Sp ,

(2.6.5)

for all j ∈ Z. Then we have new bounded-measure-theoretic structures, {(Sp , ψp,j ) : j ∈ Z} .

(2.6.6)

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 53

Proposition 2.9: Let (Sp , ψp,j ) be the measure space (2.6.6) for j ∈ Z. Then, for any generating projection αkp ∈ Sp , 1 , ∀k ∈ Z. (2.6.7) pj+1 Proof . The formula (2.6.7) is shown by Eqs. (2.6.4), (2.6.5) and (2.6.6). Now, let’s regard the p-boundary subalgebra Sp as a Banach space equipped with its C ∗ -norm (e.g., [9]). On this Banach space Sp , define bounded linear transformations cp and ap by the linear morphisms satisfying ψp,j (αkp ) = δj,k

p p cp (αkp ) = αk+1 , and ap (αkp ) = αk−1 ,

(2.6.8)

{αkp }k∈Z

for all generating projections of Sp . Then they are well-defined bounded linear transformations on Sp by Eq. (2.5.4). i.e., they are Banach-space operators acting on Sp or they are operators contained in the operator space B (Sp ) of all Banach-space operators acting on the Banach space Sp (e.g., see Ref. [9]). Definition 2.5: Let cp , ap ∈ B (Sp ) be the Banach-space operators (2.6.8). Then we call them, the p-creation, and the p-annihilation on Sp , respectively. Define a new Banach-space operator lp ∈ B (Sp ) by def

lp = cp + ap , on Sp .

(2.6.9)

Then we call lp , the p-radial operator on Sp . By Eq. (2.6.9), the iterated products lnp of the p-radial operator lp of Eq. (2.6.9) are Banach-space operators in the operator space B (Sp ) for all n ∈ N0 = N ∪ {0 }, with axiomatization: l0p = 1B(Sp ) , the identity element of B (Sp ) , and hence, one can define a subset L of B (Sp ) by def

Lp = C [{lp }] in B (Sp ),

(2.6.10)

where C [Y ] means the (pure-algebraic) polynomial algebra in a subset Y of B (Sp ) over C = C · 1B(Sp ) , and Z are the operator-norm closures of subsets Z of B (Sp ) (e.g., [9]). By definition, if L ∈ L, then ∞

L= n=0

tn lnp , with tn ∈ C.

54 ■ Constructive Analysis of Semicircular Elements

This subset L forms a Banach algebra embedded in the operator space B (Sp ), by the definition (2.6.10). Define now a unary operation (∗) on this Banach algebra L by the operation, ∗

∞ n=0

tn lnp

∞

= n=0

tn lnp in Lp ,

(2.6.11)

where tn are the conjugates of tn in C, for all n ∈ N0 . Then this unary operation (2.6.11) is a well-defined adjoint, since ∗

(L∗ ) = L, ∀L ∈ Lp ; and ∗

(L1 + L2 ) = L∗1 + L2∗ , ∀L1 , L2 ∈ Lp ; and ∗

(zL) = zL∗ , ∀z ∈ C, and L ∈ Lp ; and ∗

(L1 L2 ) = L∗2 L∗1 , ∀L1 , L2 ∈ Lp , by the very definition (2.6.11). Therefore, this Banach algebra L of Eq. (2.6.10) forms a Banach ∗-algebra in B (Sp ), equipped with the adjoint (2.6.11). So, all elements of L are adjointable (in the sense of Ref. [9]) in B (Sp ). Let Sp be the p-boundary subalgebra of Mp , and let L be the Banach ∗-algebra (2.6.10) with its adjoint (2.6.11). Define a Banach ∗-algebra LSp by the tensor product Banach ∗-algebra, def

LSp = Lp ⊗ Sp ,

(2.6.12)

where ⊗ is the tensor product of Banach ∗-algebras. Note that, since Sp is a C ∗ -algebra, it is a Banach ∗-algebra equipped with its C ∗ norm, and hence, the tensor product LSp of Eq. (2.6.12) is indeed well-defined as a Banach ∗-algebra. By Eq. (2.6.12), one can realize that the Banach ∗-algebra LSp is generated by the generating operators, uk

denote

=

lp ⊗ αkp : k ∈ Z ,

since

(2.6.13) n

unk = (lp ⊗ αkp ) = lnp ⊗αkp ,

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 55

for all n ∈ N, and, u0k = l0p ⊗ 1Sp = 1L ⊗ 1Sp = 1LSp , the unity of LSp . Now, let’s consider the p-radial operator lp ∈ L of Eq. (2.6.9) more in detail. Observe first that the p-creation cp and the p-annihilation ap satisfy that cp ap = 1B(Sp ) = ap cp , on Sp , since, for all generating projections αkp ∈ Sp , ( p ) p p cp ap (αkp ) = cp αk−1 = α(k− 1)+1 = αk , and ( p ) p p ap cp (αkp ) = ap αk+1 = α(k+1)− 1 = αk , implying the above operator-equality in B (Sp ) by Eq. (2.5.4). So, for any n1 , n2 ∈ N0 , cnp1 anp2 = anp2 cnp1 on Sp ,

(2.6.14)

with axiomatization: 0 c0p =1B(Sp ) = ap , in B (Sp ) ,

satisfying that n

n

n n n n cp ap = (cp ap ) = 1B(Sp ) = (ap cp ) = ap cp ,

(2.6.15)

for all n ∈ N0 . By Eq. (2.6.14), if lp ∈ Sp is the p-radial operator, generating Lp , then n n n n lp = (cp + ap ) = ckp an−k , p k k=0

where

(2.6.16) n k

=

n! , ∀k ≤ n ∈ N0 . k!(n − k)!

By Eqs. (2.6.15) and (2.6.16), for any n ∈ N, n 2n−1 lp

= k=0

2n − 1 k

)−k ckp a(2n−1 , p

56 ■ Constructive Analysis of Semicircular Elements

and

(2.6.17) n

l2n p

= k=0

2n k

2n−k ckp ap =

2n n

· 1B(Sp ) + [Rest].

Lemma 2.1: Let lp be the p-radial operator. If n ∈ Nis odd, then lnp does not contain 1B(Sp ) − summand. (2.6.18) If n ∈ N is even, then lnp contains the 1B(Sp ) − summand,

n n 2

· 1B(Sp ) . (2.6.19)

Proof . The statements (2.6.18) and (2.6.19) are proven by Eq. (2.6.17). The above lemma shows that the iterated product lnp ∈ L contains the identity 1Lp = 1B(Sp ) in the Banach ∗-algebra Lp , if and only if n is even in N, or n = 0 , for all n ∈ N0 . Now, define the bounded linear transformations, Ep,j : LSp → Sp by the linear morphisms satisfying ( Ep,j (uk ) =

pj+1 n 2

(2.6.20) )n+1 ( +1

) lnp (αkp ) ,

for all generating operators {uk = lp ⊗ αkp }k ∈Z ⊂ LSp of Eq. (2.6.13), for all j ∈ Z, where n2 are the minimal integers greater than or equal to n2 , for all n ∈ N; for example, 3 4 =2= , etc. 2 2 Then these morphisms {Ep,j }j∈Z of Eq. (2.6.20) are well-defined bounded linear transformations by the cyclicity (2.6.10) of the tensorfactor Lp , and the structure theorem (2.5.4) of the other tensor-factor Sp .

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 57

Define now the corresponding linear functionals {τp,j }j∈Z on LSp by def

τp,j = ψp,j ◦ Ep,j , on LSp ,

(2.6.21)

for all j ∈ Z, where {ψp,j }j∈Z are the linear functionals (2.6.5) on the p-boundary subalgebra Sp . Theorem 2.1: Let LSp be the tensor product Banach ∗-algebra (2.6.12), and let τp,j be a linear functional (2.6.21) for an arbitrary j ∈ Z. Then, for any generating element uk ∈ LSp of Eq. (2.6.13), τp,j (unk ) = δj,k ωn p2(j+1)

n 2

c n2

,

(2.6.22)

for all n ∈ N, for all k , j ∈ Z, where 1 0

ωn =

if n is even if n is odd,

and cn are the n-th Catalan numbers. Proof . Let uk = lp ⊗ αkp be a generating element (2.6.13) of LSp , for k ∈ Z. For any fixed j ∈ Z, ( ( )) ( ) τp,j (unk ) = τp,j lnp ⊗ αkp = ψp,j Ep,j lnp ⊗ αkp by Eq. (2.6.21) = ψp,j =

=

n+1 (n p ) (pj+1 ) ψ lp (αk ) p,j n ⌈ 2 ⌉+1  n+1 n (pj+1 )   ψp,j  n  ⌈ n2 ⌉+1  2

    

=

n+1 (n p ) (p j +1 ) lp (αk ) n ⌈ 2 ⌉+1

          

n+1

(pj+1 ) ⌈ n2 ⌉+1

ψp,j (No αkp -terms)

n+1

(pj+1 ) ⌈ n2 ⌉+1

δj,k ψp,j n+1

(pj+1 ) ⌈ n2 ⌉+1

αkp + [Rest]

n n 2

αjp + [Rest]

( ) δj,k ψp,j No αjp -terms

if n is even

if n is odd if n is even

if n is odd

58 ■ Constructive Analysis of Semicircular Elements

by Eq. (2.6.7)       =     

(pj+1 )

= δj,k ωn

n+1

n

δj,k

n 2 +1

( ) ψp,j αjp

n 2

if n is even

n+1

(pj+1 ) ⌈ n2 ⌉+1 (pj+1 )

δj,k (0) = 0

if n is odd

n+1

n 2 +1

n!

( n2 )!( n2 )!

1 pj+1

by Eq. (2.6.5), where ωn = 1 (resp., ωn = 0), if and only if n is even (resp, odd) in N ( )n n! = δj,k ωn pj+1 ( n2 )!( n2 +1)! ( )n ( )n = δj,k ωn pj+1 c n2 = δj,k ωn p2(j+1) 2 c n2 , where ck are the k -th Catalan numbers for all k ∈ N0 . Therefore, the formula (2.6.22) holds true. The above theorem shows that, for our Banach ∗-algebra LSp of Eq. (2.6.12), the linear functionals {τp,j }j∈Z of Eq. (2.6.21) are welldefined, and, for the generating elements {uk }k∈Z of Eq. (2.6.13), we have the statistical information (2.6.22) on LSp ; τp,j (uk ) = δj,k ωn p2(j+1)

n 2

c n2 ,

for all n ∈ N. Definition 2.6: Let LSp be the Banach ∗-algebra (2.6.12) for p ∈ P, and let τp,j be a linear functional (2.6.21) on LSp for j ∈ Z. The bounded-measure-theoretic structure, LSp (j)

denote

=

(LSp , τp,j )

(2.6.23)

is called the j -th p-adic filter.

2.7

ON THE FREE ADELIC FILTERIZATION LS

Let p ∈ P be a prime, and Sp , the p-boundary subalgebra of the p-C ∗ algebra Mp , and let LSp (j) = (LSp , τp,j )

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 59

be the j -th p-adic filter (2.6.23), for j ∈ Z. By Eq. (2.6.22), one can characterize the statistical data of the generating operators, {upk = lp ⊗ αkp : k ∈ Z} of LSp (j), by the formula, ( n) τp,j (upk ) = δj,k ωn p2(j+1)

(2.7.1) n 2

c n2 ,

(2.7.2)

for all n ∈ N, and k ∈ Z. Different from Sections 2.5 and 2.6, we denote the generating elements uk ∈ LSp , in the sense of Eq. (2.6.13) for a fixed prime p, by ukp as in Eq. (2.7.1). Since multi p-adic filters {LSp }p∈P will be used from below, we had better indicate the generating elements precisely like in Eq. (2.7.1). Recall the free-probabilistic free product (e.g., see Refs. [4,6,7,12, 13,15]). Let {LSp (j) : p ∈ P, j ∈ Z} (2.7.3) be the collection of all j -th p-adic filters LSp (j) of Eq. (2.6.23) for all p ∈ P, and j ∈ Z. From these “commutative” boundedmeasure-theoretic structures (2.7.3), define the (free-probabilistic “noncommutative”) Banach ∗-probability space, LS

denote

=

(LS, τ ) ,

by the free product, (LS, τ ) =

⋆

p∈P,j∈Z

LSp (j) ,

i.e.,

(2.7.4) LS = ⋆

p∈P

LS⋆|Z| , and τ = ⋆ p

p∈P

⋆ τp,j ,

j∈Z

in the sense of Refs. [13,15], where LSp⋆|Z| means the free product of |Z|-copies of LSp , for p ∈ P. By the definition (2.7.4), this Banach ∗-probability space LS is a huge free-probabilistic structure induced from {Qp }p∈P . Definition 2.7: We call the free product Banach ∗-probability space (2.7.4), denote LS = (LS, τ ) of the p-adic filters (2.7.3) for all p ∈ P, the free Adelic filterization.

60 ■ Constructive Analysis of Semicircular Elements

By definition, the free Adelic filterization LS is the free product Banach ∗-probability space having its countable-infinitely many free blocks LSp (j), the j -th p-adic filters, for all p ∈ P, and j ∈ Z. 2.7.1

Weighted-Semicircular Elements

In this subsection, we introduce weighted-semicircular elements in an arbitrary topological ∗-probability space (B, ψ). Let (B, ψ) be a topological ∗-probability space with a topological ∗-algebra B , and a bounded linear functional ψ on B . Definition 2.8: A self-adjoint free random variable a ∈ (B , ψ) is said to be weighted-semicircular with its weight t0 ∈ C× = C \ {0} (or, in short, t0 -semicircular), if n

ψ (an ) = ωn t02 c n2 , or, if

(2.7.5) 



knψ a, a, ..., a = δn,2 t0 , n-times

for all n ∈ N, where k•ψ (...) is the free cumulant of Ref. [13] on B in terms of ψ (under the M¨ obius inversion). By definition, one can verify that all 1 -semicircular elements are the semicircular elements of Definition 1.1. Indeed, by the M¨obius inversion of Ref. [13], if a self-adjoint free random variable a ∈ (B , ψ) satisfies the free-cumulant data in  Eq. (2.7.5),  then  ψ (an ) = π∈N C(n)

 ψ   k|V | a, a, ..., a  V ∈π



= π∈N C(n)

δ|V |,2 t0

V ∈π



= θ∈NC2 (n)

|V |-times

B∈θ

t0

= θ∈N C2 (n)

#(θ)

t0

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 61

where NC2 (n) = {θ ∈ NC (n) : ∀B ∈ θ ⇒ |B | = 2 } is the subset of the noncrossing-partition lattice NC (n), consisting of all pair partitions whose blocks have their cardinalities 2, and where #(θ) means the number of blocks in the partition θ   n

=

n

n

1 = t02 |N C2 (n)|

t02 = t02  θ∈N C2 (n)

θ∈N C2 (n)

since the numbers of the blocks in all pair partitions θ are identical to n 2 , by the very definition of NC2 (n) n

= ωn t02 N C

n 2

n

= ωn t02 c n2 ,

(2.7.6)

( ) for all n ∈ N, since NC2 (n) is equipotent (or bijective) to NC n2 , only when n ∈ N is even, and the cardinality of all noncrossing-partition lattice NC (k ) are identical to the k -th Catalan number ck (for all k ∈ N), where ωn are in the sense of Eq. (2.6.22). So, the free-moment data and the free-cumulant data provide an equivalent weighted-semicircularity by Eq. (2.7.6) as in Eq. (2.7.5). 2.7.2

Weighted-Semicircular Elements of LS

In this section, we study weighted-semicircular elements in our free Adelic filterization LS = (LS, τ ) of Eq. (2.7.4). Theorem 2.2: Let LS be the free Adelic filterization, and let ujp ∈ LSp (j ) be the “j -th” generating operator of the “j -th” p-adic filter, for j ∈ Z, and p ∈ P. Then such a free random variable ujp ∈ LSp (j) is p 2 (j +1 ) -semicircular in LS. i.e., ujp ∈ LSp (j ) =⇒ it is p 2 (j +1 ) -semicircular in LS.

(2.7.7)

Proof . Suppose ujp = lp ⊗ αjp be the j -th generating operator of the j -th p-adic filter LSp (j), which is a free block of our free Adelic filterization LS. Then it is self-adjoint in LS, since ( p )∗ ( )∗ uj = l∗p ⊗ αjp = lp ⊗ αjp = upj , in LS. Since ujp is contained in a free block LSp (j) of LS, the iterated ( )n powers upj are also contained in the same free block LSp (j), as free

62 ■ Constructive Analysis of Semicircular Elements

reduced words with their lengths-1 in LS by Eq. (2.7.4), for all n ∈ N (e.g., [13,15]). Therefore, τ

((

upj

)n )

= τp,j

(( p )n ) uj = ωn p2(j+1)

n 2

c n2 ,

for all n ∈ N, by Eq. (2.6.22). It implies that the self-adjoint free random variable ujp ∈ LSp (j) is p 2 (j +1 ) -semicircular in LS by Eq. (2.7.5). Therefore, the statement (2.7.7) holds. The following result is also obtained by Eq. (2.6.22). Theorem 2.3: Let LS be the free Adelic filterization, and let ukp ∈ LSp (j) be the k -th generating operator of the j -th p-adic filter LSp (j), a free block of LS, where k ̸= j in Z. Then the free distribution of ukp ∈ LSp (j) is the zero free distribution in LS. i.e., ukp ∈ LSp (j ) with k ̸= j =⇒ it has the zero free distribution in LS. (2.7.8) p Proof . By the self-adjointness of uk ∈ LS, the free distribution of it is characterized by the free-moment sequence, ( ( p n ))∞ τ (uk ) n=1 . n

Since the powers (upk ) are the free reduced words with their lengths-1 in LS, contained in the free block LSp (j), we have that ( ( n) n) τ (upk ) = τp,j (upk ) = 0, by Eq. (2.6.22) because k ̸= j in Z, for all n ∈ N. i.e., the free distribution of ukp is characterized by the free-moment sequence, (0, 0, 0, 0, . . .) , implying that the free distribution of it is the zero free distribution in LS. Therefore, the statement (2.7.8) holds.

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 63

By Eqs. (2.7.7) and (2.7.8), one can verify that lots of free random variables in the free Adelic filterization LS have the zero free distribution. i.e., in a free block LSp (j) of LS, the only “j -th” generating element ujp provides nonzero free-distributional data in LS by Eq. (2.7.7), for all j ∈ Z and p ∈ P. Now, define a new free random variable Ujp ∈ LSp (j ) in LS by def

Ujp =

1 pj+1

upj ∈ LSp (j) ⊂ LS,

(2.7.9)

for all j ∈ Z, and p ∈ P, where ujp are the j -th generating elements of the free block LSp (j) of LS, which are p 2 (j +1 ) -semicircular in LS, by Eq. (2.7.7). Theorem 2.4: Let Ujp ∈ LSp (j ) be the free random variables (2.7.9) of the free Adelic filterization LS, for all j ∈ Z and p ∈ P. Then they are semicircular in LS. i.e., Ujp =

1 p j +1

ujp ∈ LSp (j ) =⇒ it is semicircular in LS.

(2.7.10)

Proof . Observe first that such a free random variable Ujp ∈ LSp (j ) is self-adjoint in LS, since (

Ujp

)∗

1

=

(

pj+1

upj

)∗

=

1 pj +1

upj = Ujp ,

1 in LS, by the self-adjointness of ujk in LS, and by the fact that pj+1 ∈ × R in C . ( ) n Since Ujp are the free reduced words with their lengths-1 in LS, contained in the same free block LSp (j), we have

τ

((

Ujp

)n )

= τp,j =

(( p )n ) Uj = τp,j 1

p j +1

n

τp,j

1

(

pn(j+1)

(( p )n ) uj =

upj

)n

1 p n(j +1 )

ωn p n(j +1 ) c n2

by Eq. (2.6.22) = ωn c n2 , for all n ∈ N. Therefore, this self-adjoint free random variable Ujp ∈ LSp (j ) is semicircular in LS by Eq. (1.2.2). i.e., the statement (2.7.10) holds true.

64 ■ Constructive Analysis of Semicircular Elements

The above theorem concludes that the p-adic number fields {Qp }p∈P induces the Banach ∗-probability space LS, the free Adelic filterization, and generate the semicircular elements, Ujk =

1 pj+1

(lp ⊗ αkp ) ∈ LSp (j) : j ∈ Z, p ∈ P ,

in LS. Equivalently, our analysis of p-adic number fields {Qp }p∈P let us have semicircular elements in the free Adelic filterization LS. 2.7.3

The Adelic Semicircular Filterization LS

As we have discussed in Section 2.7.2, “many” free random variables of the free Adelic filterization LS have the zero free distribution by Eq. (2.7.8). So, let’s focus on free random variables whose free distributions are nonvanishing dictated by Eqs. (2.7.7) and (2.7.10) in LS. Define two families U and U of LS by U = upj ∈ LSp (j) : j ∈ Z, p ∈ P , and

(2.7.11) U =

Ujp =

1 p p u : uj ∈ U pj+1 j

,

where {LSp (j) : j ∈ Z, p ∈ P} are the free blocks (2.7.3) of LS. Definition 2.9: Let (B, ψ) be a topological ∗-probability space, and let S ⊂ B be a subset. The family S is said to be a free (weighted-) semicircular family of (B , ψ), if (i) S is a free family consisting of mutually free elements of (B , ψ), and (ii) all elements of S are (weighted-)semicircular in (B, ψ). Theorem 2.5: Let U and U be the subsets (2.7.11) of the free Adelic filterization LS. The family Uis a free weighted-semicircular family of LS. (2.7.12)

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 65

The family U is a free semicircular family of LS.

(2.7.13)

Proof . Let U = upj ∈ LSp (j) : j ∈ Z, p ∈ P be the family of Eq. (2.7.11) in LS. Since all elements ujp are chosen from the mutually distinct free blocks LSp (j) of Eq. (2.7.3), they are mutually free from each other in LS by Eq. (2.7.4), and hence, this family U is a free family of LS. Furthermore, by Eq. (2.7.7), all elements ujp ∈ U are p 2 (j +1 ) -semicircular in LS. Therefore, the family U is a free weighted-semicircular family of LS. Let U = Ujp ∈ LSp (j) : j ∈ Z, p ∈ P be the family of Eq. (2.7.11) in LS. By the freeness on U, this family U is also a free family. Moreover, by Eq. (2.7.10), all elements of U are semicircular in LS. Therefore, the family U is a free semicircular family of LS by Definition 2.9. Note that, by Eqs. (2.7.7), (2.7.8) and (2.7.10), the free family U, or U of Eq. (2.7.11) generates free reduced words of LS in U, respectively, in U having “non-zero” free distributions in LS. So, to concentrate on studying free random variables of LS with nonzero free distributions, we need to focus on the generating family U or U of LS. Proposition 2.10: Let L1 = C [U], and L2 =C [U ] be the Banach ∗-subalgebras of the free Adelic filterization LS. Then L1 = L2 in LS,

(2.7.14)

set-theoretically. Proof . The set-equality (2.7.14) is shown by the definition (2.7.11). Indeed, one has upj = pj+1 Ujp ∈ U ⇐⇒ Ujp =

1 pj+1

ujp ∈ U ,

for all j ∈ Z, p ∈ P. So, L1 = C [U] = C [U ] = L2 , in LS. So, the set-equality (2.7.14) holds.

66 ■ Constructive Analysis of Semicircular Elements

The above proposition shows that the Banach ∗-subalgebras L1 and L2 are actually identical from each other in LS by Eq. (2.7.14). Definition 2.10: Let U be a free semicircular family (2.7.13), and let LS = C [U ] = C [U], in LS, where U is the free weighted-semicircular family (2.7.12). Then the free-probabilistic sub-structure, LS

denote

=

(LS, τ = τ |LS )

(2.7.15)

of LS is called the (free-)semicircular Adelic filterization (of LS). As we discussed above, if W ∈ LS is a free reduced word in U , then it has nonzero free distribution in LS, and conversely; if W ∈ LS is a free reduced word having its nonzero free distribution, then it is a word in U (or in U), contained in the semicircular Adelic filterization LS (e.g., see (1.4.13)). Theorem 2.6: Let LS be the semicircular Adelic filterization (2.7.15). Then ∗-iso

LS =

⋆

p∈P,j ∈Z

C

[

Ujp

] ∗-iso = C

⋆

p∈P,j∈Z

Ujp

,

(2.7.16)

in the free Adelic filterization LS, where (⋆) in the first ∗-isomorphic relation of (2.7.16) is the free-probabilistic free product of Refs. [13,15], and (⋆) in the second ∗-isomorphic relation of (2.7.16) is the pure-algebraic free product inducing noncommutative free words in ∪ Ujp = U . p∈P,j∈Z

Proof . Let LS be the semicircular Adelic filterization generated by the free semicircular family U of Eq. (2.7.13). Since the generating family U is a free family, the first ∗-isomorphic relation of Eq. (2.7.16) is immediately obtained. Note that all free (reduced or non-reduced) words in U have their unique free reduced words of LS under the operator-multiplication on LS by Eq. (2.7.4), and hence, the second ∗-isomorphic relation of Eq. (2.7.16) holds, too.

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 67

The above theorem allows us to analyze the free-distributional information on the semicircular Adelic filterization LS by Eq. (1.4.13). Corollary 2.5: Let Ujp11 , ..., UjpNN ∈ U be the generating semicircular elements of LS, for N ∈ N (without considering repetition), and N let W = Ujpl l ∈ LS be a free random variable. Then the freel=1

distributional data τ (W ) is characterized by Eq. (1.4.13). Proof . The proof is done by Eqs. (1.4.13) and (2.7.16). The above corollary confirms that the universality of the semicircular law lets us characterize the general free-probabilistic information on the semicircular Adelic filterization LS by the joint free moment formula (1.4.13).

2.8

CIRCULAR ELEMENTS OF LS

In this section, we consider circular elements of the semicircular Adelic filterization LS of Eq. (2.7.15). Corollary 2.6: Suppose (p1 , j1 ) ̸= (p2 , j2 ) in the Cartesian product set P × Z. Then the free random variable √ ) 1 ( T = √ Ujp11 + iUjp22 ∈ LS, with i = −1, 2 is a circular element, whose free distribution is characterized by the only nonzero joint free cumulants (1.3.8) or by the only nonzero joint free moments (1.3.13) of {T, T ∗ }. Proof . By assumption, two semicircular elements Ujp11 and Ujp22 are distinct semicircular elements of the generator set U , and hence, they are free in LS, by Eq. (2.7.16). So, the free random variable T is a well-defined circular element of LS by Eq. (1.3.1). Therefore, by the universality of the circular law, the free distribution of T ∈ LS is characterized by the only nonvanishing joint free cumulants, k2 (T ∗ , T ) = 1 = k2 (T, T ∗ ) ,

68 ■ Constructive Analysis of Semicircular Elements

by Eq. (1.3.8), where k• (...) is the free cumulant on LS in terms of τ , equivalently, by the only nonvanishing joint free moments, n

n

τ ((T ∗ T ) ) = |X2 (2n)| = τ ((T T ∗ ) ) , for all n ∈ N by Eq. (1.3.13), where X2 (2n) is the sets (1.3.11).

2.9

FREE POISSON ELEMENTS OF LS INDUCED BY U ∪ U

Let U be the free weighted-semicircular family (2.7.12), and U , the free semicircular family (2.7.13) of the free Adelic filterization LS, generating the semicircular Adelic filterization LS of Eq. (2.7.15). In this section, we study certain free Poisson elements of LS induced by U or by U . Corollary 2.7: Let (p1 , j1 ) ̸= (p2 , j2 ) in the Cartesian product set P × Z, and let Ul = Ujpl l ∈ U be the corresponding generating semicircular elements (2.7.9) of LS, for l = 1 , 2 . Then a self-adjoint free random variable, W = U1 U2 U1 ∈ LS is a free Poisson element, whose free distribution is characterized by kn (W, ..., W ) = ωn c n2 , or, by

(2.9.1) 



τ (W n ) = ωn  π∈N Ce (n)

c |V | V ∈π

2

,

for all n ∈ N, where NCe (n) is a subset of the noncrossing-partition lattice NC (n), N Ce (n) = {θ ∈ N C(n) : ∀B ∈ θ, |B| is even in N} . Proof . By assumption, two free random variables U1 and U2 are distinct semicircular elements in the free family U , and hence, they are free in LS. By the self-adjointness of U2 , the free random variable W = U1 U2 U1 is indeed a well-defined free Poisson element of LS by

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 69

Eq. (1.3.14). Thus, the free distribution of W is characterized by the free moments, |V |

τ (W n ) =

τ U2 π∈N C(n)

,

V ∈π

by Eq. (1.3.18) or by the free cumulants, kn (W, ..., W ) = τ (U2n ) , by Eq. (1.3.17), for all n ∈ N. Therefore, the free Poisson distribution of W is characterized by Eq. (2.9.1), by the semicircularity of U2 ∈ U in LS. Also, one can get the following result. Corollary 2.8: Suppose (p1 , j1 ) ̸= (p2 , j2 ) in P × Z, and let U1 = Ujp11 ∈ U , and u2 = ujp22 ∈ U in LS. Then a free random variable W = U1 u2 U1 ∈ LS is a free Poisson element of LS, whose free distribution is characterized by n(j +1) kn (W, ..., W ) = ωn p2 2 c n2 , or, by

(2.9.2) 



n(j2 +1)

τ (W n ) = ωn p2

c |V |

 π∈N Ce (n)

V ∈π

,

2

for all n ∈ N. 2 (j +1 )

Proof . Since u2 ∈ U is p2 2 -semicircular in LS, it is self-adjoint in LS. And, by assumption, two free random variables u2 = pj22 +1 U2 and U1 are free in LS. Therefore, the free random variable W = U1 u2 U1 is a well-defined free Poisson element of LS by Eq. (1.3.14). So, the free Poisson distribution of W is characterized by the free moments, |V |

τ (W n ) =

τ u2 π∈N C(n)

V ∈π

,

70 ■ Constructive Analysis of Semicircular Elements

by Eq. (1.3.18) or, by the free cumulants, n(j2 +1)

kn (W, ..., W ) = τ (un2 ) = ωn p2

c n2 ,

by Eq. (1.3.17), implying that 

 |V |(j2 +1)

τ (W n ) = ωn 

p2

c |V | 2

V ∈π

π∈N Ce (n)

,

respectively, n(j2 +1)

kn (W, ..., W ) = ωn p2

c n2 ,

for all n ∈ N. Remark that, for any π ∈ N C(n), for all n ∈ N, |V | = n. V ∈π

Therefore, the free-distributional data (2.9.2) holds. Theorem 2.7: Let (p1 , j1 ) = ̸ (p2 , j2 ) in the Cartesian product set P × Z, and let u1 = ujp11 ∈ U , and U2 = Ujp22 ∈ U in LS. Then a free random variable W = u1 U2 u1 ∈ LS is a free Poisson element of LS, whose free distribution is characterized by 2n(j +1) kn (W, ..., W ) = ωn p1 1 c n2 , or, by

(2.9.3) 



2n(j1 +1)

τ (W n ) = ωn p1

c |V |

 π∈N Ce (n)

V ∈π

2

,

for all n ∈ N. Proof . Let W = u1 U2 u1 ∈ LS be given as above. Then, by the selfadjointness of u1 = pj11 +1 U1 and U2 , the free random variable W ∈ LS is self-adjoint. Observe that W = p1j1 +1 Ujp11 U2 pj11 +1 Ujp11 ,

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 71

satisfying 2(j1 +1)

W = U 1 p1

U2 U1 in LS, 2 (j +1 )

where U1 = Ujp11 ∈ U . Remark that an operator p1 1 U2 ∈ LS is self-adjoint, and it is free from U1 in LS. Therefore, this self-adjoint free random variable W is a free Poisson element of LS, by Eq. (1.3.14). Therefore, the free distribution of W is characterized by the free moments, |V | 2(j +1) τ (W n ) = τ p1 1 U2 π∈N Ce (n)

V ∈π

2n(j1 +1 )



= ωn p1

π∈NCe (n)

c |V |

,

2

V ∈π

by Eq. (1.3.18), or, equivalently, by the free cumulants, kn (W , ..., W ) = τ

2 (j1 +1 )

p1

n

U2

2n(j1 +1 )

= p1

(

) ωn c n2 ,

by Eq. (1.3.17), for all n ∈ N. Therefore, the free-distributional data (2.9.3) holds. Also, we obtain the following generalized result. Theorem 2.8: Suppose (p1 , j1 ) ̸= (p2 , j2 ) in P × Z, and let ul = ujpl l ∈ U be the corresponding weighted-semicircular elements of LS, for l = 1 , 2 . Then a free random variable W = u1 u2 u1 ∈ LS is a free Poisson element, whose free distribution is characterized by 4(j1 +1) 2(j2 +1) p2

kn (W, ..., W ) = δn,2 p1

,

or, equivalently, by

(2.9.4)  2(j1 +1) j2 +1 p2

τ (W n ) = ωn p1

c |V |

 π∈N Ce (n)

for all n ∈ N.



n V ∈π

2

,

72 ■ Constructive Analysis of Semicircular Elements

Proof . By assumption two free random variables u1 and u2 are distinct in the free weighted-semicircular family U, and hence, they are free in LS. Observe that 2(j1 +1) j2 +1 p 2 U2

W = u1 u2 u1 = U1 p1

U1 ,

2 (j +1 )

in LS. If we let T = p1 1 p2j2 +1 U2 , then it is not only self-adjoint in LS but also free from the semicircular element U1 ∈ U in LS. So, by Eq. (1.3.14), this operator W is a free Poisson element of LS, and hence, the corresponding free Poisson distribution of W is characterized by  2(j1 +1) j2 +1 p2

τ (W n ) = ωn 

p1

π∈N Ce (n)



|V |

c |V |

V ∈π

2

,

by Eq. (1.3.18), or, by 2n(j1 +1) n(j2 +1) p2

kn (W, ..., W ) = p1

( ) ωn c n2 ,

by Eq. (1.3.17), for all n ∈ N, implying the free-distributional data (2.9.4).

2.10

BANACH-SPACE OPERATORS ACTING ON LS

In this section, we consider certain bounded linear transformations (or Banach-space operators) acting on our semicircular Adelic filterization LS, by regarding this Banach ∗-algebra as a Banach space (e.g., [9]). Let B (LS) be the operator space consisting of all bounded linear transformations on the Banach space LS. From now on, let’s regard the set P of all primes as a totally ordered set (or simply, a TOset), {q1 < q2 < q2 < · · · } , where (≤) is the total ordering on P (inherited from that on N), i.e., q1 = 2, q2 = 3, q3 = 5, q5 = 11, etc.

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 73

For an arbitrary prime p ∈ P, we write p+1 = the very next prime under ordering, and

(2.10.1) p−1 = the very previous prime (if exists),

up to the total ordering on P, for instance, 11+1 = 13, 13+1 = 17, 3−1 = 2, 11−1 = 7, and 2−1 is undefined in P because 2 is the smallest prime. By extending the above notation (2.10.1), one can naturally define the following terminology; for any p ∈ P, p+k =

( ) (p+1 )+1 +1...

(k-times), +1

respectively,

(2.10.2)

p−k =

( ) (p−1 )−1 −1...

−1

(k-times, if exists),

for all k ∈ N. For example, 3+3 = 11 , 13−2 = 7 , 5−2 = 2 , and 3−3 is undefined in P, etc. In the rest of this section, we use the notations (2.10.1) and (2.10.2). Remark that a prime p ∈ P is the np -th prime in the TOset P, i.e., qnp = p in (P, ≤), for instance, n3 = 2, n11 = 5, n23 = 9, etc., saying that: 3 is the 2-nd prime; 11 is the 5-th prime and 23 is the 9-th prime in P, respectively. It is easily verified that if p ∈ P is the np -th prime, then p−np is undefined in P, and

(2.10.3) p−np +1 = 2, the smallest prime in P.

74 ■ Constructive Analysis of Semicircular Elements

Notation 2.2: Throughout this section, we use the notations (2.10.1), (2.10.2) and (2.10.3). For convenience, we denote the Cartesian product set P × Z by P, i.e., denote P = P × Z.

✥

On the set P, define functions s+1 , s−1 : P → P by s+1 (p, j) = (p+1 , j + 1) , and

(2.10.4) s−1 (p, j) =

  (p−1 , j − 1) 

if p−1 exists

Undefined

otherwise,

for all (p, j) ∈ P, where p±1 are in the sense of Eqs. (2.10.1) and (2.10.2). By extending the definition (2.10.4), define functions, s+k , s−k : P → P, by s+k (p, j) = (p+k , j + k) , and

(2.10.5) s−k (p, j) = (p−k , j − k) ,

for all k ∈ N0 = N ∪ {0 }, with axiomatization: s±0 = idP , the identity map on P. Definition 2.11: Let {s±k }k∈N0 be the functions (2.10.5) on P. Then they are called the prime-integer shiftings (in short, pi-shiftings) on P. Note again that s−np (p, j) is undefined in P, and

(2.10.6) s−np +1 (p, j) = (2, j − np + 1) , in P,

by Eq. (2.10.3).

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 75

From the pi-shiftings, we now define “multiplicative” linear morphisms, σ±1 : LS → LS, by the linear morphisms satisfying  p±1 if p±1 are well-defined ( )  Uj±1 σ±1 Ujp =  O otherwise, for all Ujp ∈ U , where O is the zero element of LS, and U is the free semicircular family (2.7.13) generating the semicircular Adelic filterization LS (e.g., [4]). From below, if we simply write ( ) p±1 “σ±1 Ujp = Uj±1 ,” (2.10.7) for the conditional meaning of Eq. (2.10.7). N ( )n If W = Ujpl l l ∈ LS is a free reduced word with its length-N , l=1

for n1 , ..., nN ∈ N, then N (p )

l ±1 Ujl ±1

σ±1 (W ) =

nl

∈ LS,

l=1

(in the sense of Eq. (2.10.7)’), by the multiplicativity of the morphisms σ±1 of Eq. (2.10.7). So, these morphisms are indeed well-defined multiplicative linear transformations on LS. Consider that ( )∗ ( ) ( p±1 )∗ ( )∗ p±1 σ±1 tUjp = σ±1 tUjp = tUj±1 = t Uj±1 = σ±1 tUjp , for all t ∈ C, and Ujp ∈ U , implying that ∗

σ±1 (T ∗ ) = σ±1 (T ) , in LS,

(2.10.8)

for all T ∈ LS, by Eq. (2.7.16). The relation (2.10.8) shows that our multiplicative bounded linear transformations σ±1 of Eq. (2.10.7) are ∗-homomorphisms on LS. Let Hom (LS) = {β : LS → LS : β is a ∗-homomorphism} be the homomorphism semigroup on LS.

76 ■ Constructive Analysis of Semicircular Elements

Lemma 2.2: A multiplicative linear transformations σ±1 of Eq. (2.10.7) are ∗-homomorphisms on LS, i.e., σ±1 ∈ Hom (LS).

(2.10.9)

Proof . By definition, the linear morphisms σ±1 are bounded. Moreover, under their multiplicativity, they are ∗-homomorphisms on LS by Eq. (2.10.8). It is not hard to check that the ∗-homomorphisms σ±1 are induced by the pi-shiftings s±1 of Eq. (2.10.4) acting on the index set P of the generator set U of the semicircular Adelic filterization LS. So, generally, let’s define the ∗-homomorphisms {σ±k }k∈N0 on LS by def

k

σ±k = (σ±1 ) , ∀k ∈ N0 , with identities:

(2.10.10) 0

σ±0 = (σ±1 ) = 1LS , the identity map of Hom (LS) . Since σ±1 ∈ Hom (LS), σ±k ∈ Hom (LS), by Eqs. (2.10.9) and (2.10.10). Definition 2.12: The ∗-homomorphisms σk ∈ Hom (LS) of k ∈ Z,  if k ≥ 0  σ+k σk =  σ−|k| if k < 0, in the sense of Eq. (2.10.10) are called the k -th prime-integer shifts (in short, the k -th pi-shifts) on LS. Lemma 2.3: Let σk ∈ Hom (LS) be the k -th pi-shifts (2.10.10) on LS, for k ∈ Z. Then ( ) σ−np Ujp = O, the zero element of LS, and

(2.10.11) p)

2 σ−(np −1) Uj = Uj−n , in LS, p +1

(

where p ∈ P is the np -th prime in (P, ≤), as in Notation 2.2. Furthermore, if a free reduced word W ∈ LS contains its free-factor Ujp ∈ U , then

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 77

σ−np (W ) = O, in LS.

(2.10.12)

Proof . The formulas of Eq. (2.10.11) are shown by Eqs. (2.10.6) and (2.10.10). The formula (2.10.12) is proven by Eqs. (2.10.7) and (2.10.10). By the above lemma, one obtains the following generalized result. Proposition 2.11: Let σk ∈ Hom (LS) be the k -th pi-shifts on LS, and Ujp ∈ U , a generating semicircular element of LS, where p ∈ P is the np -th prime in the TOset (P, ≤). If n ≥ np in N, then ( ) σ−n Ujp = O, in LS, and hence, if W ∈ LS is a free reduced word containing a free-factor Ujp ∈ U and if n ≥ np , then σ−n (W ) = O, in LS.

(2.10.13)

Proof . The proof of the formula (2.10.13) is done by Eqs. (2.10.11) and (2.10.12). By Eq. (2.10.13), one can verify the following free-distributional data affected by the action of the pi-shifts on the semicircular Adelic filterization LS. Theorem 2.9: Let {σk }k∈Z ⊂ Hom (LS) be the k -th pi-shifts on LS. If k ≥ 0 , then ( ) σk Ujp ∈ U , too, ∀Ujp ∈ U , in LS. If k < 0, with k = − |k |, then ( ) σk Ujp ∈ U in LS ⇐⇒ |k| < np , in N0 ,

(2.10.14)

where np is the order of a prime p in the TOset (P, ≤); and ( ) σk Ujp = O, ⇐⇒ |k| ≥ np . (2.10.15) Proof . The relations (2.10.14) and (2.10.15) are shown by Eq. (2.10.13).

78 ■ Constructive Analysis of Semicircular Elements

By definition, if σk ∈ Hom (LS) is the k -th pi-shift on LS, with k ≥ 0 , then ( ) p+k σk Ujp = Uj+k ∈ U , in LS, and hence, the semicircularity of Ujp ∈ U is preserved to be the p

+k semicircularity of Uj +k ∈ U in LS, whenever k ≥ 0 in Z.

The following corollary is immediately obtained by Theorem 2.9. Corollary 2.9: Let Ujp ∈ U be a generating semicircular element of LS, and let σk ∈ Hom (LS) be the k -th pi-shift on LS, for k ∈ Z. ( ) Suppose k ≥ 0 . Thenσk Ujp ∈ U is semicircular in LS. (2.10.16) ( ) Suppose k = − |k | < 0 . Then σk Ujp ∈ U is semicircular in LS, ( ) if and only if |k | < np in N; meanwhile, σk Ujp = O in LS, if and only if |k| ≥ np in N.

(2.10.17)

✥

By Eqs. (2.10.16) and (2.10.17), we obtain the following freeprobabilistic information on LS affected by the action of pi-shifts. Theorem 2.10: Let W =

N (

l=1

Ujpl l

)n l

∈ LS be a free reduced word with

its length-N in the free semicircular family U , for n1 , ..., nN ∈ N, for N ∈ N. If σk ∈ Hom (LS) is the k -th pi-shift on LS, for k ∈ Z. If k ≥ 0 , then τ (W ) = τ (σk (W )) , which is characterized by Corollary 2.6 or (1.4.13).

(2.10.18)

If k < 0 , with |k| < npl , where npl are the orders of the primes pl in the TOset (P, ≤) , for all l = 1 , ..., N , then,

(2.10.19)

τ (W ) = τ (σk (W )) , characterized by Corollary 2.6 or (1.4.13). If k < 0 , with |k| ≥ npl , for at least one l ∈ {1 , ..., N }, then (2.10.20) τ (W ) = ̸ τ (σk (W )) = 0, in general.

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 79

Proof . Note that if nonzero, then the free reduced words W and σk (W ) have the same noncrossing-partition-structures as we considered in Section 1.4. So, by the semicircularity, if σk (W ) is nonzero, τ (W ) = τ (σk (W )) , in C, characterized by Corollary 2.6 or (1.4.13). Therefore, statements (2.10.18) and (2.10.19) hold true. If it is identical to the zero element of LS, equivalently, if there exists at least one free factor Ujpl l of W satisfies (2.10.13), then τ (W ) = ̸ τ (O) = 0, because τ (W ) = ̸ 0, in general, by Corollary 2.6 or (1.4.13). So, the statement (2.10.20) holds. The above theorem characterizes how our pi-shifts affect the original free-distributional data on the semicircular Adelic filterization LS.

2.11

Pi-SHIFT OPERATORS ON LS

Let LS = (LS, τ ) be the semicircular Adelic filterization, generated by the free semicircular family, U =

Ujp =

1 p u : p ∈ P, j ∈ Z , pj+1 j

where ujp = lp ⊗ αjp ∈ U are the p 2 (j +1 ) -semicircular elements of the free Adelic filterization LS, for all p ∈ P, and j ∈ Z. Consider that, since all ∗-homomorphisms on LS are bounded (multiplicative) linear transformations on LS, they are well-defined Banach-space operator acting on the Banach space LS, i.e., they are elements of the operator space B (LS) (e.g., [9]), equivalently, Hom (LS) ⊂ B (LS) . Thus, our pi-shifts {σk }k∈Z are understood as Banach-space operators on LS, where  if k ≥ 0  σ+k σk =  σ−|k| if k < 0,

80 ■ Constructive Analysis of Semicircular Elements

where {σ±n }n∈N0 are in the sense of Eq. (2.10.10). Recall that if np is the order of a prime p in the TOset (P, ≤), then  p+k if k ≥ 0  Uj+k    ( )  p−|k| Uj −|k| if k < 0, and |k| < np σk Ujp = (2.11.1)      O if k < 0, and |k| ≥ np , in LS, by Eqs. (2.10.16) and (2.10.17), satisfying  if either k ≥ 0,   (( p )n )   n = τ or ω c U  n j 2 (( ( ))n ) k < 0, |k| < np τ σk Ujp =      0 otherwise,

(2.11.2)

for all n ∈ N, by Eqs. (2.10.18), (2.10.19) and (2.10.20). i.e., as Banachspace operators of B (LS), the pi-shifts {σk }k∈Z satisfy (2.11.1) and (2.11.2). In particular, the free-distributional data (2.11.2) illustrate that the action of pi-shifts on LS deforms the original free probability on LS, and the deformation is characterized by the very data. Define a subset S of B (LS) by L = {σk }k∈Z in B (LS) ,

(2.11.3)

and we call this set S of Eq. (2.11.3), the pi-shift family on LS (or, of B (LS)). Observe now that, if k1 , k2 ≥ 0 in Z, then, for any generating semicircular element Ujp ∈ U of LS, ( ) p+k2 σk1 σk2 Ujp = σk1 Uj+k 2

( ) p+(k1 +k2 ) (p+k2 )+k1 = Uj+k = Uj+(k = σk1 +k2 Ujp , 2 +k1 1 +k2 )

implying that k1 , k2 ≥ 0 =⇒ σk1 σk2 = σk1 +k2 in S ,

(2.11.4)

in LS, by Eq. (2.11.1). Now, assume that k1 , k2 < 0 in Z. Then, for any generating element Ujp ∈ U of LS,

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 81

( ) σk1 σk2 Ujp =

=

 p−|k |   σk1 Uj −|k22 |

if |k2 | < np

 

if |k2 | ≥ np

σk1 (O) = O

 p−|k2 |−|k1 |  Uj−|k2 |−|k1 | 

=

O  p−|k1 +k2 |  Uj −|k1 +k2 | 

= σk1 +k2

O ( p) Uj ,

if |k1 | < np−|k2 | otherwise if |k1 + k2 | < np otherwise

implying that k1 , k2 < 0 =⇒ σk1 σk2 = σk1 +k2 in S ,

(2.11.5)

where S is the pi-shift family (2.11.3) of B (LS). However, different from (2.11.4) and (2.11.5), one obtains the following result. Suppose k > 0 in Z, and take σk ∈ S , and Ujq ∈ U , where q is the k -th prime in P, i.e., nq = k in N. Then ( ) q+k q+k−k σ−k σk Ujq = σ−k Uj+k = Uj+k−k = Ujq , meanwhile,

(2.11.6) ( ) q−k σk σ−k Ujq = σk Uj−k = σk (O) = O,

in LS, by Eq. (2.11.1). The above relation (2.11.6) shows that, in general, σk σ−k = ̸ σ−k σk , in “B (LS) , ” and hence,

(2.11.7) σk1 σk2 = ̸ σk2 σk1 , in B (LS) ,

in general, implying that σk 1 σk 2 ∈ / S , in B (LS) ,

(2.11.8)

82 ■ Constructive Analysis of Semicircular Elements

in general, for k1 , k2 ∈ Z, where S is the pi-shift family (2.11.3). In particular, one obtains the following result. Proposition 2.12: Let S = {σk }k∈Z be the pi-shift family, and let k1 , k2 ∈ Z be arbitrary. If either k1 = 0 , or k2 = 0 , then σk1 σk2 = σk1 +k2 = σk2 σk1 in S . (2.11.9) If either k1 , k2 ≥ 0 , or k1 , k2 < 0 , then σk1 σk2 = σk1 +k2 = σk2 σk1 in S . (2.11.10) In general, σk1 σk2 ∈ / S in B (LS), and σk1 σk2 = ̸ σk2 σk1 in B (LS), i.e., σk1 σk2 = ̸ σk1 +k2 , and σk2 σk1 = ̸ σk1 +k2 ,

(2.11.11)

in general, in B (LS). Proof . The proofs of the statements (2.11.9) and (2.11.10) are done by Eqs. (2.11.4) and (2.11.5), respectively. The statement (2.11.11) is shown by Eqs. (2.11.6), (2.11.7) and (2.11.8). The above proposition, especially (2.11.11), shows that the pi-shift family S of Eq. (2.11.3) does not have a nice algebraic structure in B (LS) (or, in Hom (LS)). However, one can construct a (closed) subspace S of the operator space B (LS) by def

S = C [S ] in B (LS),

(2.11.12)

where C [S ] is the polynomial algebra in the pi-shift family S over C = C · I, where I is the identity operator of B (LS), and Y are the operator-norm closures of subsets Y of B (LS). Remark that, since Hom (LS) ⊂ B (LS) , the (pure-algebraic polynomial) algebra C [S ] (in S ) is well-defined in B (LS) as a (pure-algebraic) subspace, and hence, its topological closure S of Eq. (2.11.12) is a well-defined closed subspace of B (LS). i.e., since the products of pi-shifts are again well-defined Banach-space

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 83

operators on LS, the subset S of Eq. (2.11.12) is well-defined in B (LS). Moreover, by definition, it forms a Banach algebra embedded in B (LS). By definition, if T ∈ S, then it is expressed by   ∞

T =

n

n=0

σkl  ∈ S,

t(k1 ,...,kn )

 (k1 ,...,kn

)∈Zn

l=1

with t(k1 ,...,kn ) ∈ C, by Eq. (2.11.12). Definition 2.13: Let S be the Banach algebra (2.11.12) embedded in the operator space B (LS) on the semicircular Adelic filterization LS. We call S the pi-shift-operator algebra (acting on LS), and all elements of S are said to be pi-shift operators (on LS). In the rest of this section, we focus on the pi-shift operators T ∈ S formed by n T = σkl ∈ S, (2.11.13) l=1

where k1 , ..., kn are not necessarily distinct in Z, for pi-shifts σkl ∈ S , for all l = 1 , ..., n, for n ∈ N. Theorem 2.11: Let T ∈ S be a pi-shift operator (2.11.13), and let Ujp ∈ U be a generating semicircular element of LS. Then  p+(k +k if p+(ki +,,,+kn ) exist, ...+k1 )   U nn n−1   in P, ∀i = 1, 2, ..., n ( ) j+ kl l=1 T Ujp =     O otherwise, in LS, and hence,

(2.11.14)

(( ( ))n ) τ T Ujp =

    ωn c n2   

O

if p+(ki +,,,+kn ) exist, in P, ∀i = 0, 1, ..., n − 1 otherwise,

for all n ∈ N, with the identity k0 = 0 in Eq. (2.11.14). n Proof . Suppose T = σkl ∈ S be a given pi-shift operator. Then, l=1

for a generating semicircular element Ujp ∈ U or LS,

84 ■ Constructive Analysis of Semicircular Elements

( ) T Ujp =

n

σkl

(

) Ujp =

l=1

  q   U 

n

j+

kl

if p+(ki +,,,+kn ) exist, in P, ∀i = 0, 1, ..., n − 1

l=1

   

O

otherwise, (2.11.15)

by Eqs. (2.11.1), (2.11.9), (2.11.10) and (2.11.11), where

q = p+(k1 +...+kn ) =

(p+kn )+kn−1

, +kn−2 ...

+k1

in P (if exists), for k1 , ..., kn ∈ Z . Therefore, the operator-equality in Eq. (2.11.14) holds by Eq. (2.11.15). ( ) By Eq. (2.11.15), one can realize that T Ujp ∈ U in LS, if the primes p+(ki +...+kn ) are well-defined in P, for all i = 0 , 1 , ..., n − 1 , where k0 is assumed to be 0 ; it becomes the zero free random variable O of LS, if there exists at least one is ∈ {i1 , ..., in } such that p+(kis +kis+1 +...+kn ) is undefined in P. ( ) So, if T Ujp ̸= O in LS, then it is contained in the free semicircular family U , generating LS, and hence, it is semicircular in LS. So,

τ

(( ( p ))n ) T Uj = ωn c n2 , ∀n ∈ N.

( ) Of course, if T Ujp = O in LS, then the free distribution of it is the zero free distribution in LS. Therefore, the free-distributional data (2.11.14) hold. The above theorem characterizes how the pi-shift operators (2.11.13) formed by the iterated products of pi-shifts deform the semicircular law for the generating semicircular elements of U in LS. This characterization (2.11.14) illustrates how the pi-shift operators of S deform the free probability on LS by Corollary 2.6 (or (1.4.13)), and (2.11.12).

Semicircular Elements Induced by p-Adic Number Fields Qp ■ 85

REFERENCES [1] I. Cho, Free Semicircular Families in Free Product Banach *-Algebras Induced by p-Adic Number Fields Over Primes p, Compl. Anal. Oper. Theory, 11, no. 3, (2017) 507–565. [2] I. Cho, Certain *-Homomorphisms Acting on Unital C*-Probability Spaces and Semicircular Elements Induced by p-Adic Number Fields over Primes p, Elect. Res. Arch., 28, no. 2, (2020) 739–776. [3] I. Cho, On Dynamical Systems Induced by p-Adic Number Fields, Opuscula Math., 35, no. 4, (2015) 445–484. [4] I. Cho, Banach-Space Operators Acting on Semicircular Elements Induced by p-Adic Number Fields over Primes p, Mathematics, 7, (2019) 498. DOI: 10.3390/math7060498. [5] I. Cho, Asymptotic Semicircular Laws Induced by p-Adic Number Fields over Primes p, Compl. Anal. Oper. Theory, 13, no. 7, (2019) 3169–3206. [6] I. Cho, and T. Gillespie, Free Probability on the Hecke Algebra, Compl. Anal. Oper. Theory, (2014). DOI: 10.1007/s11785-014-0403-1. [7] I. Cho, and P. E. T. Jorgensen, Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields over Prime p, Math. Spec. Issue Math. Phys. II, (2020). DOI:10.3390/math7020190. [8] I. Cho, and P. E. T. Jorgensen, Semicircular Elements Induced by p-Adic Number Fields, Opuscula Math., 35, no. 5, (2017) 665–703. [9] A. Connes, Noncommutative Geometry. Academic Press: San Diego, CA (1994). ISBN: 0-12-185860-X. [10] T. Gillespie, Superposition of Zeroes of Automorphic L-Functions and Functoriality, PhD Thesis, CERTAIN FREE SEMICIRCULAR FAMILIES 47, University of Iowa, (2010). [11] T. Gillespie, Prime Number Theorems for Rankin-Selberg L-Functions over Number Fields, Sci. China Math., 54, no. 1, (2011) 35–46. [12] F. Radulescu, Random Matrices, Amalgamated Free Products and Subfactors of the C*-Algebra of a Free Group of Nonsingular Index, Invent. Math., 115, (1994) 347–389. [13] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Am. Math. Soc. Mem., 132, no. 627, (1998) x+88. CRM Monograph Series, vol 1., (1992)

86 ■ Constructive Analysis of Semicircular Elements [14] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet & East European Mathematics, vol 1. World Scientific: Singapore (1994). ISBN: 978-981-02-0880-6 [15] D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, vol 1. American Mathematical Society: Ann Arbor, MI (1992). ISBN-13: 978-0821811405.

CHAPTER

3

Semicircular Elements Induced by Orthogonal Projections

I

n chapter 2, we studied how p-adic number fields {Qp }p∈P induce the semicircular elements, { } U = Ujp ∈ LSp (j) : p ∈ P, j ∈ Z ,

generating the Banach ∗-probability space (LS, τ ), the semicircular Adelic filterization. And, from a certain shifting process on the Cartesian product set P × Z, we defined the pi-shift family S = {σk }k∈Z in the homomorphism semigroup Hom (LS), and constructed the corresponding Banach algebra S embedded in the operator space B (LS). In particular, it was considered how the pi-shift operators of S deform the original free probability on LS. For more about the motivations for this chapter, see [1–4]. And, see [15] for fundamental operator-theoretic background. In this section, motivated by Chapter 2, we extend or generalize the constructions of semicircular elements from a family of mutually orthogonal |Z|many projections in a C ∗ -probability space and then consider the counterparts of the main results of Chapter 2, under our generalized settings. There are various different approaches to construct semicircular elements (e.g., [5,8,10,26,27]). They generally fall into two groups: (i) analysis on measure spaces (i.e., an approach from classical statistics) DOI: 10.1201/9781003374817-3

87

88 ■ Constructive Analysis of Semicircular Elements

and approaches used in (ii) topological *-probability spaces (including that of C ∗ -probability spaces, or W ∗ -probability spaces, e.g., see Refs. [16–19,25]). Our approach in this section is different because they universalize those of Chapter 2. That is, we here offer a new construction of semicircular elements. It is motivated by a systematic study of “weighted-semicircular” elements as in Chapter 2 (e.g., also, see Refs. [9,12]) under abstract conditions. In conclusion, the construction of this section is different from, and independent of, those in earlier works. Also, see Refs. [10,11] for additional discussion. In Chapter 2, we applied number-theoretic results (e.g., [14,23]), and free-probabilistic techniques (e.g., [6–8,17,18,20]) to consider freeprobabilistic models, and we realized there are well-defined (weighted-) semicircular elements. Motivated by the main results of Chapter 2, in Ref. [10], the similar constructions of (weighted-)semicircular elements were considered from “arbitrary” C ∗ -probability spaces containing |Z|-many mutually orthogonal projections. The main results of Ref. [10] showed that whenever one can have mutually orthogonal |Z|-many projections in a C ∗ -probability space, the corresponding weighted-semicircular elements whose weights are characterized by the free-distributional data of the projections; moreover, under suitable (additional) conditions, semicircular elements are well-defined (See below). As an application of Ref. [10], the author and Jorgensen considered free, semicircular elements induced by orthogonal projections acting on infinite-dimensional separable Hilbert spaces in Ref. [11]. It shows that whenever there are countable (finitely or infinitely) many orthogonal projections, one can have the corresponding countable free semicircular family.

3.1

SOME BANACH ∗-ALGEBRAS INDUCED BY PROJECTIONS

In this section, we fix a C ∗ -probability space (A, ψ) containing its unity (or the multiplication identity) 1A of a C ∗ -algebra A, with either ψ (1A ) < ∞, or ψ (1A ) = ∞. (i.e., we allow a case where ψ is unbounded for the C ∗ -topology on A.) Remark that, as usual (e.g., [19,21,22,24,25]), we may assume this C ∗ -probability space is unital in the sense that: ψ (1A ) = 1,

Semicircular Elements Induced by Orthogonal Projections ■ 89

however, we will not assume the unitality of (A, ψ) because our theory is (not-exactly-same-but) motivated by Chapter 2. Recall that each p p-C ∗ -algebra Mp contains its unity αQ , satisfying ( p) 1 1 φp,j αQ = j − j+1 ̸= 1, ∀p ∈ P, j ∈ Z. p p Assume that there are |Z|-many projections {qj }j∈Z in the C ∗ -algebra A, i.e., the operators qj satisfy qj∗ = qj = qj2 in A, with |ψ (qj )| < ∞ , for all j ∈ Z (e.g., [14]), and qj = 1A j∈Z

Assume further that they are mutually orthogonal in A, in the sense that: qj1 qj2 = δj1 ,j2 qj1 in A, for all j1 , j2 ∈ Z. Now, we fix the family, Q = {qj ∈ A : mutually orthogonal}j∈Z .

(3.1.1)

Remark 3.1: One can have such a C ∗ -algebraic structure A containing a family Q of Eq. (3.1.1), naturally or artificially. Clearly, in the settings of Chapter 2 (also, see e.g., [5,9,10,12]), one can naturally take such structures. Suppose now there is a C ∗ -algebra A0 containing a family, QN = {q1 , ..., qN } , for some N ∈ N∞ , of mutually orthogonal N -many projections q1 , ..., qN , where N∞ = N∪ {∞}. Here, ∞ is the countable infinity χ0 = |N|. Then, under suitable direct product, or tensor product, or free product of copies of A0 , under product topology, one can construct a C ∗ -algebra A containing a family Q of |Z|-many mutually orthogonal projections, containing QN , under unitary equivalence.

90 ■ Constructive Analysis of Semicircular Elements

For instance, if a C ∗ -algebra A0 contains a family QN of N -many mutually orthogonal projections q1 , ..., qN , for N < ∞ in N, then one can construct a new C ∗ -algebra A, the direct product C ∗ -algebra, ⊕|Z|

A = A0

∞

= ⊕

n=−∞

A0,n , with A0,n = A0 .

having its family, ∞

Q= ∪

n=−∞

QN,n , with QN,n = QN ⊂ A0,n = A0 ,

of mutually orthogonal |Z|-many projections. So, QN is embedded infinitely many times in Q. ∞ Also, if A0 contains a family Q∞ = {qj }j=1 , then one may take a C ∗ -algebra A, whose Banach-space expression is as follows: A = A0 ⊕ (A0 ⊖ C) , where A0 ⊖ C is the orthogonal complement of C in A0 (as a Banach space, where ⊕ means the direct product of Banach spaces), having its family, Q = {. . . , x−2 , x−1 , x0 , x1 , x2 , . . .} , where xn =

  qn+1

if n ≥ 0

q|n|+1

if n < 0,



(as in Ref. [11]), for all n ∈ Z. i.e., whenever a C ∗ -algebra A0 contains countably many mutually orthogonal projections, one can have a C ∗ -algebra A containing |Z|-many mutually orthogonal projections containing the original projections of A0 , artificially-but-canonically.

✥

Define now a C ∗ -subalgebra Q of A by def

Q = C ∗ (Q) = C [Q] in A,

(3.1.2)

where Q = {qj }j∈Z is the family (3.1.1), and where C ∗ (Y ) mean the C ∗ -subalgebras of A generated by the subsets Y ⊆ A, and Z are the C ∗ -topology closures of subsets Z of A.

Semicircular Elements Induced by Orthogonal Projections ■ 91

Proposition 3.1: Let Q be the C ∗ -subalgebra (3.1.2) of a fixed C ∗ algebra A, generated by the family Q of Eq. (3.1.1). Then ∗-iso

∗-iso

Q = ⊕ (C · qj ) = C⊕|Z| , in A.

(3.1.3)

j∈Z

Proof . Since the family Q = {qj }j∈Z of Eq. (3.1.1) is a collection of projections, to show the relation (3.1.3), it is enough to check the orthogonality on Q. Define now linear functionals {ψj }j ∈Z on the C ∗ -algebra Q of Eq. (3.1.2) by the linear morphisms satisfying that ψj (qk ) = δj,k ψ (qj ) , for all k ∈ Z,

(3.1.4)

for all j ∈ Z, where ψ is the linear functional of the fixed C ∗ -probability space (A, ψ). (Note that, even though ψ is (bounded or) unbounded, the linear functionals ψj of Eq. (3.1.4) are bounded!) The bounded linear functionals {ψj }j∈Z of Eq. (3.1.4) are welldefined on Q by the structure theorem (3.1.3). Assumption 3.1: Let (A, ψ) be a fixed C ∗ -probability space, and let Q be the C ∗ -subalgebra (3.1.2) of A. In the rest of this section, we assume that ψ (qj ) ∈ C× = C \ {0} , ∀j ∈ Z.

✥

By the well-defined linear functionals of Eq. (3.1.4) on Q, one can have the corresponding free-probabilistic structures, {(Q, ψj ) : j ∈ Z},

(3.1.5)

in (A, ψ). Definition 3.1: Let (Q, ψj ) be a C ∗ -probability space (3.1.5) for j ∈ Z, embedded in (A, ψ). We call (Q, ψj ), the j -th filter of Q (in (A, ψ)). Now, let’s regard the C ∗ -algebra Q of Eq. (3.1.2) as a Banach space and define the linear transformations c and a acting on the Banach space Q by the linear morphisms satisfying c (qj ) = qj+1 , and a (qj ) = qj−1 ,

(3.1.6)

92 ■ Constructive Analysis of Semicircular Elements

for all j ∈ Z. Then these linear transformations of Eq. (3.1.6) are welldefined (bounded) operators on Q, by Eq. (3.1.3). i.e., they are welldefined Banach-space operators (in the sense of Ref. [13]) acting on Q. i.e., if B (Q) is an operator space (e.g., [13]) consisting of all Banachspace operators (i.e., bounded linear transformations) on (the Banach space) Q, then c, a ∈ B (Q) . Definition 3.2: The Banach-space operators c and a on Q are called the creation, and the annihilation, respectively. Define a new Banachspace operator l on Q by l = c + a,

on Q.

(3.1.7)

Then we call l ∈ B (Q), the radial operator on Q. Note that, by the definition (3.1.6) and (3.1.7), the powers ln of the radial operator l are contained in B (Q), too, for all n ∈ N0 , with axiomatization: l0 = 1B(Q) , the identity operator on Q, i.e., ln ∈ B (Q), for all n ∈ N0 . Now, define a (closed) subspace L of B (Q) by def

L = C [{l}], in B (Q),

(3.1.8)

where Y are the operator-norm closures of subsets Y of B (Q). Recall that the operator norm ∥.∥ on B (Q) is defined by ∥T ∥ = sup ∥T (h)∥Q : h ∈ Q, ∥h∥Q = 1 , for all T ∈ B (Q), where ∥.∥Q is the C ∗ -norm on Q (inherited by that on A). Indeed, since {ln }n∈N0 ⊂ B (Q), the (pure-algebraic) polynomial algebra C [{l}] is a well-defined subset of B (Q), and hence, the subspace L of Eq. (3.1.8) is well-established in B (Q), as a Banach algebra embedded in B (Q). So, if T ∈ L, then ∞

tk lk , with tk ∈ C.

T = k=0

Semicircular Elements Induced by Orthogonal Projections ■ 93

On this Banach algebra L of Eq. (3.1.8), define a unary operation (∗) by ∗

∞

tk l

k

∞

=

k=0

tk lk in L,

(3.1.9)

k=0

where tk are the conjugates of tk in C, for all k ∈ N0 . Proposition 3.2: Every operator of the Banach algebra L is adjointable (in the sense of Ref. [13]) in B (Q). Equivalently, the closed subspace L of Eq. (3.1.8) is a Banach ∗-algebra, embedded in the operator space B (Q), equipped with its adjoint (3.1.9). Proof . It is sufficient to show that the operation (3.1.9) is a welldefined adjoint on the Banach algebra L. By definition, ∗

T ∗∗ = T , and (zT ) = z T ∗ , ∀z ∈ C, T ∈ L; and ∗

(T1 + T2 ) = T1∗ + T2∗ , ∀T1 , T2 ∈ L; and ∗

(T1 T2 ) = T2∗ T1∗ , ∀T1 , T2 ∈ L. Therefore, the operation (3.1.9) is a well-defined adjoint on L. i.e., L is a Banach ∗-algebra in B (Q). Equivalently, all elements T ∈ L have its unique adjoint T ∗ ∈ L, and hence, they are adjointable in the sense of Ref. [13] on Q. Let L be the Banach ∗-algebra (3.1.8) generated by the radial operator l of Eq. (3.1.7). Define a tensor product Banach ∗-algebra LQ by LQ = L ⊗ Q,

(3.1.10)

where ⊗ is the tensor product of Banach ∗-algebras. Since L is a Banach ∗-algebra (by Proposition 3.2), and Q is a ∗ C -algebra, the tensor product Banach ∗-algebra LQ of Eq. (3.1.10) is well-defined. Definition 3.3: The Banach ∗-algebra LQ of Eq. (3.1.10) is said to be the radial-projection (Banach ∗-)algebra (on Q).

94 ■ Constructive Analysis of Semicircular Elements

By definition, the radial-projection algebra LQ is generated by the elements, denote uj = l ⊗ qj , ∀j ∈ Z, by the cyclicity (3.1.8) of the tensor factor L, and the structure theorem (3.1.3) of the other tensor factor Q. Indeed, unj = ln ⊗ qjn = ln ⊗ qj , for all j ∈ Z, for all n ∈ N0 , with identity: u0j = l0 ⊗ qj0 = 1B(Q) ⊗ 1Q = 1L ⊗ 1, in LQ .

3.2

WEIGHTED-SEMICIRCULAR ELEMENTS INDUCED BY Q

Let (A, ψ) be a fixed C ∗ -probability space containing the family Q = {qj }j ∈Z of mutually orthogonal projections satisfying Assumption 3.1, and Q = C ∗ (Q), the corresponding C ∗ -subalgebra of A, inducing the filters, {(Q, ψj ) : j ∈ Z} , and let LQ be the radial-projection algebra (3.1.10) on Q. We here construct weighted-semicircular elements induced by the family Q. Throughout this section, we denote the generating operators of LQ by uk = l ⊗ qk ∈ LQ , (3.2.1) for all k ∈ Z. Before proceeding, let’s consider how the creation c and the annihilation a acting on Q. Observe first that ac (qj ) = a (qj+1 ) = q(j+1)−1 = qj , and

(3.2.2) ca (qj ) = c (qj−1 ) = q(j−1)+1 = qj ,

for all generating projections qj ∈ Q of Q, i.e., ca = 1Q = ac, on Q.

Semicircular Elements Induced by Orthogonal Projections ■ 95

Lemma 3.1: Let c and a be the creation, respectively, the annihilation of Eq. (3.1.6) on Q. Then n

n

cn an = (ca) = 1Q = (ac) = an cn , and

(3.2.3) cn1 an2 = an2 cn1 , on Q,

for all n1 , n2 , n ∈ N0 , with axiomatization: c0 = 1Q = a0 , on Q. Proof . By Eq. (3.1.3), the formulas of Eq. (3.2.2) imply that ca = 1Q = ac, on Q. Thus, the first formulas of Eq. (3.2.3) hold for all n ∈ N0 , and hence, the second formulas of Eq. (3.2.3) automatically hold. By Eq. (3.2.3), one can verify that, on Q, the radial operator l of Eq. (3.1.7) satisfies that n

ln = (c + a) =

n

k=0

n k

ck an−k ,

(3.2.4)

for all n ∈ N, where n k

=

n! , ∀k ≤ n ∈ N0 . k!(n − k)!

By Eq. (3.2.4), for any k ∈ N, we have 2k−1

l2k−1 = i=0

2k − 1 i

ci a2k−1−i ,

and

(3.2.5) 2k 2k

l

= i=0

on Q.

2k i

c2k a2k−i =

2k k

1Q + [Rest terms],

96 ■ Constructive Analysis of Semicircular Elements

Lemma 3.2: Let l be the radial operator (3.1.7) on Q. (I) If n ∈ N is odd, then ln does not contain the 1Q -summand. n (II) If n ∈ N is even, then ln contains the 1Q -summand, ·1Q . n 2

Proof . The proofs of the statements (I) and (II) are done by the straightforward computations (3.2.5) by Eq. (3.2.4). By Lemma 3.2, one can realize that, if qj ∈ Q is a generating projection of Q, then  n n   if n is odd ck an−k (qj )   k  k=0 ln (qj ) = (3.2.6)   n    qj + [Rest terms] (qj ) if n is even, n 2

for all n ∈ N. On the radial-projection algebra LQ , define linear transformations, Ej : LQ → Q, by the linear morphisms satisfying

(3.2.7) n+1

Ej (unk )

n

= E (l ⊗ qk ) =

ψ (qj ) [n] 2 +1

(ln (qk )) ,

for all j , k ∈ Z, for all n ∈ N, where ψ (qj ) ∈ C× (by Assumption 3.1) is ∗ the free ψ) ⊃ Q, [ n ]distribution of qj in our fixed C -probability space (A, and 2 are the minimal integers greater than or equal to n2 . Remark that, by the cyclicity (3.1.8) of the tensor factor L, and the structure theorem (3.1.3) of the other tensor factor Q, the linear transformations {Ej }j ∈Z of Eq. (3.2.7) are well-defined from LQ onto Q. i.e., they are bounded, surjective Banach-space operators, contained in the operator space B (LQ , Q), consisting of all operators from LQ to Q (e.g., [13]). Now, define the linear functionals {τj }j∈Z on the radial-projection algebra LQ by τj = ψj ◦ Ej , for all j ∈ Z,

(3.2.8)

Semicircular Elements Induced by Orthogonal Projections ■ 97

on LQ , where {ψj }j∈Z are filtered linear functionals (3.1.4) on Q, and {Ej }j ∈Z are the well-defined Banach-space operators (3.2.7), and where (◦) is the usual functional composition. By the linearity of ψj and Ej , indeed, the linear functional τj of Eq. (3.2.8) are well-defined on the radial-projection algebra LQ , for all j ∈ Z. Consider that if uk = l ⊗ qk is the k -th generating operator of LQ , for k ∈ Z, and if τj is the linear functional (3.2.8) for j ∈ Z, then ψ (qj )n+1 ⌈ n2 ⌉+1

τj (unk ) = τj (ln ⊗ qk ) = ψj

=

          

by Eq. (3.2.6)       =     

ψ(qj )n+1 ⌈ n2 ⌉+1 ψ(qj )n+1 n 2 +1

ψj

ψ(qj )n+1 ⌈ n2 ⌉+1 ψ(qj )n+1 n 2 +1

ψj

ln (qk )

ψj (ln (qk )) n n 2

qk + [Rest] (qk )

·0 n n 2

if n is odd

if n is even

if n is odd

qk

if n is even

by Eqs. (3.1.4), (3.2.6) and (3.2.7)  0 if n is odd    =  ψ(qj )n+1 n!  ψj (qk ) if n is even  n ( n2 )!( n2 )! 2 +1  0 if n is odd    =  n+1 n!  δ ψ (q ) if n is even  ψ (qj ) ( n2 )!( n2 +1)! j,k j j by Eq. (3.1.4)   = 

0

if n is odd

) n ( δj,k (ψ (qj ) ) c n2

if n is even,

(3.2.9)

98 ■ Constructive Analysis of Semicircular Elements

= δj ,k ωn ψ (qj )

n 2

2

c n2 ,

since ψ (qj ) = ψj (qj ) , ∀j ∈ Z, where ωn =

  1

if n is even

0

if n is odd,

 for all n ∈ N.

Lemma 3.3: Let uk = l ⊗ qk be the k -th generating operator of the radial-projection algebra LQ , for k ∈ Z, and let τj be the linear functional (3.2.8) on LQ , for j ∈ Z. Then τj (unk )

2

= δj,k ωn ψ (qj )

n 2

c n2 , ∀n ∈ N.

(3.2.10)

Proof . The formula (3.2.10) is proven by Eq. (3.2.9). Let LQ be the radial-projection algebra, and let {τj }j ∈Z be the linear functionals (3.2.8) on LQ . Then one obtains the system, LQ (j)

denote

=

(LQ , τj ) : j ∈ Z

(3.2.11)

of “commutative” Banach ∗-probabilistic structures (or measuretheoretic-or-statistical structures), since the radial-projection algebra LQ is commutative over C. Definition 3.4: The structure LQ (j) = (LQ , τj ) of Eq. (3.2.11) is called the j -th filterization, for all j ∈ Z. From the system (3.2.11) of the j -th filterizations, we define the “noncommutative” Banach ∗-probability space LQ (Z) by the free product Banach ∗-probability space, LQ ( Z )

denote

=

(LQ (Z) , τ ) = ⋆ LQ (j) , j∈Z

i.e., ⋆|Z|

LQ (Z) = ⋆ LQ = LQ , j∈Z

Semicircular Elements Induced by Orthogonal Projections ■ 99

and

(3.2.12) τ = ⋆ τj , j∈Z

in the sense of Refs. [21,22,24,25]. i.e., this Banach ∗-probability space LQ (Z) of Eq. (3.2.12) is the free product noncommutative ∗-probability space, having its free blocks {LQ (j)}j ∈Z , the j -th filterizations. Definition 3.5: The Banach ∗-probability space LQ (Z) of Eq. (3.2.12) is called the free filterization on Q. On the free filterization LQ (Z), we obtain the following freeprobabilistic information. Theorem 3.1: Let LQ (Z) be the free filterization (3.2.12). Suppose uj ∈ LQ (j ) is the “j -th” generating operator of the “j -th” filterization LQ (j), a free block of LQ (Z), for an arbitrarily fixed j ∈ Z. Then this 2 free random variable uj ∈ LQ (j ) is ψ (qj ) -semicircular in LQ (Z). i.e., 2

uj ∈ LQ (j) is ψ (qj ) -semicircular in LQ (Z), ∀j ∈ Z.

(3.2.13)

Proof . Let uj = l ⊗ qj be the j -th generating operator of the j th filterization LQ (j) = (LQ , τj ), which is a free block of the free filterization LQ (Z). Then, by the self-adjointness of uj in LQ (j), it is self-adjoint in LQ (Z); and the powers ujn are again contained in the free block LQ (j), as free reduced words of LQ (Z) with their lengths-1, by (3.2.12), for all n ∈ N. Thus, n ( ) ( ) 2 2 τ unj = τj unj = ωn ψ (qj ) c n2 , for all n ∈ N, by Eq. (3.2.10). Therefore, by Eq. (2.7.5), this self2 adjoint free random variable uj ∈ LQ (j ) is ψ (qj ) -semicircular in the free filterization QL(Z), equivalently, the statement (3.2.13) holds. The above theorem demonstrates that if there are mutually orthogonal |Z|-many projections, whose free distributions are nonzero in a C ∗ -probability space, then the corresponding weighted-semicircular elements are constructed in the free filterization induced by the projections. Theorem 3.2: Let uk ∈ LQ (j ) be the k -th generating operator of the j -th filterization LQ (j), a free block of the free filterization LQ (Z),

100 ■ Constructive Analysis of Semicircular Elements

with k ̸= j in Z. Then the free distribution of uk is the zero free distribution on LQ (Z). Proof . Assume that k = ̸ j in Z. If uk ∈ LQ (j) is a given free random variable of LQ (Z), then it is self-adjoint. So, the free distribution of it is fully characterized by the free-moment sequence, ∞

(τ (unk ))n=1 . Note that the powers ukn are the free reduced words of LQ (Z) with their lengths-1 by Eq. (3.2.12), contained in the same free block LQ (j). So, τ (unk ) = τj (unk ) = 0, ∀n ∈ N, by Eq. (3.2.10), since δj,k = 0. i.e., the free-moment sequence is identified with the zero-sequence, implying that the free distribution of uk ∈ LQ (j ) is the zero free distribution on LQ (Z), whenever k ̸= j in Z. The above two theorems fully characterize the free-probabilistic information of all generating operators of the free blocks of the free filterization LQ (Z). Define now the subset U of LQ (Z) by def

U = {uj ∈ LQ (j) : j ∈ Z}.

(3.2.14)

Corollary 3.1: The set U of (3.2.14) is a free weighted-semicircular family in the free filterization LQ (Z). Proof . Since all elements uj of the family U are taken from the mutually distinct free blocks LQ (j) of LQ (Z), the subset U is a free family by Eq. (3.2.12). Moreover, by Theorems 3.1 and 3.2 2 (in particular, by Eq. (3.2.13)), every element uj ∈ U is ψ (qj ) semicircular in LQ (Z), for all j ∈ Z. Therefore, U is a free weightedsemicircular family in LQ (Z). Observe now that the free reduced words of LQ (Z) in the free weighted-semicircular family U have nonzero free distributions in LQ (Z) by Eqs. (1.4.13), (3.2.10), (3.2.12) and (3.2.13). Equivalently, all other free reduced words of LQ (Z) “not” in U have zero free

Semicircular Elements Induced by Orthogonal Projections ■ 101

distributions in LQ (Z) by Eq. (3.2.12) and Theorem 3.2. i.e., all free reduced words of LQ (Z) whose free distributions are nonzero are the free reduced words in U! So, we will restrict our interests to the free random variables of LQ (Z) generated by the free weighted-semicircular family U. Definition 3.6: Let U be the free weighted-semicircular family (3.2.14) of the free filterization LQ (Z). The Banach ∗-subalgebra LQ generated by U, def LQ = C [U], or the free-probabilistic sub-structure, ) denote ( LQ = LQ , τ = τ |LQ ,

(3.2.15)

is called the (weighted-)semicircular (free sub-)filterization of LQ (Z). As we discussed above, all elements of the semicircular filterization LQ of Eq. (3.2.15) have “possible” nonzero free distributions in LQ (Z), equivalently, if a free random variable T has its nonzero free distribution in LQ (Z), then it is contained in LQ . 2

Theorem 3.3: Let uj ∈ U be the generating ψ (qj ) -semicircular element of the semicircular filterization LQ of Eq. (3.2.15), for j ∈ Z. If ψ (qj ) ∈ R in C× , (3.2.16) then the free random variable 1 1 def Uj = uj = (l ⊗ qj ) ψ (qj ) ψ (qj )

(3.2.17)

is semicircular in LQ . Proof . By Assumption 3.1, the free distributions ψ (qj ) of all projections qj are assumed to be in C× , and hence, the free random variables Uj of Eq. (3.2.17) are well-defined. i.e., since the quantity 1 × ψ(qj ) is well-determined in C , the corresponding free random variable Uj of Eq. (3.2.17) is well-defined in LQ . If the condition (3.2.16) holds for j ∈ Z, then Uj∗ =

1 ψ (qj )

u∗j =

1 ψ (qj )

in LQ , and hence, it is self-adjoint in LQ .

uj = Uj ,

102 ■ Constructive Analysis of Semicircular Elements

Consider now that 1 ψ (qj )

( ) τ Ujn =

n

n

τ

(

unj

)

ωn (ψ (qj ) ) c n2 = = ωn c n2 , n (ψ (qj ) )

for all n ∈ N. Thus, this self-adjoint free random variable Uj of Eq. (3.2.17) is semicircular in LQ by Eq. (1.2.2), under the condition (3.2.16). The above theorem provides a condition (3.2.16) to construct semicircular elements from our weighted-semicircular generating elements of the semicircular filterization LQ . For convenience, we assume the following condition at the end of this monograph. Assumption 3.2: Let Q = {qj }j∈Z be the family (3.1.1) of our fixed C ∗ -probability space (A, ψ). For convenience, we assume automatically that ψ (qj ) ∈ R× = R \ {0} , in C, for “all” j ∈ Z.

✥

Proposition 3.3: Let LQ be the semicircular filterization (3.2.15) generated by the free weighted-semicircular family U of Eq. (3.2.14). Let U = Uj =

1 ψ(qj ) uj

: uj ∈ U

(3.2.18)

be a subset of LQ , under Assumption 3.2. The set U of Eq. (3.2.18) is a free semicircular family in LQ .(3.2.19) LQ = C [U ], set-theoretically, in LQ (Z).(3.2.20) Proof . By Theorem 3.3, under Assumption 3.2, the family U of Eq. (3.2.18) is a free semicircular family. So, the statement (3.2.19) holds true. Consider that 1 uj = ψ (qj ) Uj ∈ U ⇐⇒ Uj = uj ∈ U , ψ (qj ) for all j ∈ Z, where U is the free weighted-semicircular family (3.2.14), generating LQ . Therefore, def

LQ = C [U] = C [U ],

Semicircular Elements Induced by Orthogonal Projections ■ 103

set-theoretically, in the free filterization LQ (Z). So, the set-equality (3.2.20) holds, too. Remark 3.2: (1) Readers can compare the construction of the semicircular Adelic filterization LS of Eq. (2.7.15), and that of the semicircular filterization LQ . (2) Readers also can realize why we simply call LQ (generated by the free “weighted-semicircular” family U), the “semicircular” filterization, by Eq. (3.2.20). From below, without loss of generality, we understand LQ as LQ = C [U ], in LQ (Z) , by Eq. (3.2.20), where U is the free semicircular family (3.2.18) under Assumption 3.2. Theorem 3.4: Let LQ be the semicircular filterization. Then ∗-iso

LQ =

∗-iso

⋆ C [{Uj }] = C

j∈Z

⋆ {Uj } ,

j∈Z

(3.2.21)

1 in LQ (Z), where Uj = ψ(q uj ∈ U are the semicircular elements j) (3.2.17) (under Assumption 3.2), for all j ∈ Z, and where (⋆) in the first ∗-isomorphic relation of Eq. (3.2.21) is the free-probabilistic free product of Refs. [21,22,24,25], and (⋆) in the second ∗-isomorphic relation of Eq. (3.2.21) is the algebraic free product inducing the noncommutative free words in ∪ {Uj } = U . j∈Z

Proof . By Eqs. (3.2.15), (3.2.19) and (3.2.20), the first ∗-isomorphic relation of Eq. (3.2.21) holds in the semicircular filterization LQ (Z). All free words in U have their unique free-reduced-word-forms (under the operator-multiplication) as operators of LQ . Therefore, the second ∗-isomorphic relation of Eq. (3.2.21) holds, too.

3.3

FREE-DISTRIBUTIONAL DATA ON LQ

Let LQ = (LQ , τ ) be the semicircular filterization (3.2.15) generated by the free weighted-semicircular family U of Eq. (3.2.14), or, by the free semicircular family U of Eq. (3.2.18) (under Assumption 3.2),

104 ■ Constructive Analysis of Semicircular Elements

satisfying the structure theorem (3.2.21) in the free filterization LQ (Z) on Q ⊂ (A, ψ). In Section 3.2, we showed that n ( ) 2 2 τ unj = ωn ψ (qj ) c n2 , ∀uj ∈ U, and

(3.3.1) ( ) τ Ujn = ωn c n2 , ∀Uj ∈ U , in LQ .

Theorem 3.5: Let Ujl ∈ U be the generating semicircular elements of LQ , for l = 1 , ..., N , for N ∈ N, where j1 , ..., jN are not necessarily N

distinct in Z. If W =

Ujl is a free random variable of LQ , then the

l=1

free-distributional data τ (W ) is characterized by Eq. (1.4.13) for an N -tuple (j1 , ..., jN ) ∈ ZN . Proof . By the universality (1.2.4), or (1.2.5) of the semicircular law, and by the structure theorem (3.2.21) of LQ , the joint free moment τ (W ) is computed by the formula (1.4.13), since each free factors Ujl of W satisfies (3.3.1). The above theorem verifies that the combinatorial computation technique (1.4.13) is well-applicable to consider free-distributional data on our semicircular filterization LQ , under the universality of the semicircular law. 2

Theorem 3.6: Let ujl ∈ U be the generating ψ (qjl ) -semicircular elements of LQ , for l = 1 , ..., N , for N ∈ N, where j1 , ..., jN are not N

necessarily distinct in Z. Suppose W =

Ujl is a free random variable l=1

of LQ generated by the generating semicircular elements Ujl ∈ U , for the same indices j1 , ..., jN ∈ Z. If N

ujl ∈ LQ ,

w= l=1

then

(3.3.2) N

τ (w) = τ (W )

ψ (qjl ) , l=1

where τ (W ) is characterized by Theorem 3.5.

Semicircular Elements Induced by Orthogonal Projections ■ 105

Proof . Under hypothesis, by Theorem 3.5, the joint free moment ϖ = τ (W ) is characterized by Eq. (1.4.13). Observe that N Y

τ (w) = τ

!

!

ψ (qjl ) W

=ϖ

l=1

N Y

! ψ (qjl ) ,

l=1

since ujl = ψ (qjl ) Ujl , ∀l = 1 , ..., N , and hence, w=

N Y l=1

! ψ (qjl )

N Y

! Ujl

l=1

=

N Y

! ψ (qjl ) W,

l=1

in LQ , implying the free-distributional data (3.3.2).

3.4

P

CIRCULAR ELEMENTS OF LQ

In Sections 3.1 and 3.2, we constructed semicircular elements,   1 U = Uj = (l ⊗ qj ) ∈ LQ (j) : j ∈ Z , ψ (qj ) generating the semicircular filterization LQ , from a fixed family, Q = {qj }j∈Z ⊂ (A, ψ) of mutually orthogonal |Z|-many projections whose free distributions are ψ (qj ) ∈ R× , ∀j ∈ Z, and, in Section 3.3, we characterize the free-distributional data (of free reduced words) on LQ (in the generating family U ∪ U ). As application, in this section, we consider circular elements of LQ . Theorem 3.7: Let j1 ̸= j2 in Z, and let Uj1 , Uj2 ∈ U be the generating semicircular elements of the semicircular filterization LQ . Then a free random variable, 1 T = √ (Uj1 + iUj2 ) ∈ LQ 2

106 ■ Constructive Analysis of Semicircular Elements

is a circular element. The circular law for T is characterized by the only nonzero joint free cumulants (1.3.8); k2 (T, T ∗ ) = 1 = k2 (T ∗ , T ) , where k• (...) is the free cumulant on LQ in terms of τ ; or the only nonzero joint free moments (1.3.13); n

n

τ ((T ∗ T ) ) = |X2 (2n)| = τ ((T T ∗ ) ) , for all n ∈ N. Proof . Since j1 ̸= j2 , the chosen semicircular elements Uj1 and Uj2 are distinct in generating free semicircular family U of LQ , implying the freeness of them in LQ . Thus, by Eq. (1.3.1), the free random variable T is a circular element in LQ . By the universality of the circular law, this circular element T has the only nonzero joint free cumulants, k2 (T, T ∗ ) = 1 = k2 (T ∗ , T ) , by Eq. (1.3.8), and the only nonzero joint free moments, n

n

τ ((T ∗ T ) ) = |X2 (2n)| = τ ((T T ∗ ) ) , by Eq. (1.3.13).

3.5

FREE POISSON ELEMENTS OF LQ INDUCED BY U ∪ U

In this section, we consider some free Poisson elements of the semicircular filterization LQ induced by the free weighted-semicircular family U of Eq. (3.2.14), or the free semicircular family U of Eq. (3.2.18), generating LQ . Theorem 3.8: Fix j ∈ Z, and let T ∈ LQ be a nonzero self-adjoint free random variable, satisfying T ∈

⋆

C [{Uk }] \ C, in LQ .

k∈Z\{j}

Then a new free random variable, W = Uj T Uj ∈ LQ

Semicircular Elements Induced by Orthogonal Projections ■ 107

is a free Poisson element. The free Poisson distribution of W is characterized by the free moments,   τ T |V |

τ (W n ) = ωn  π∈N Ce (n)

or by the free cumulants, 

,

V ∈π

(3.5.1) 

kn W, W, ..., W  = τ (T n ) , n-times

for all n ∈ N. Proof . Under hypothesis, a self-adjoint free random variable W ∈ LQ is a free Poisson element by Eq. (1.3.14) because Uj and T are free in LQ . Thus, the free-distributional data (3.5.1) characterize the free distribution of W , by Eqs. (1.3.17) and (1.3.18). By the above general result, one obtains the following corollary. Corollary 3.2: Let j1 = ̸ j2 in Z, and let Uj1 , Uj2 ∈ U be the corresponding generating semicircular elements of LQ . Then the free random variable, W = Uj1 Uj2 Uj1 ∈ LQ is a free Poisson element, whose free distribution is characterized by the free moments,   τ (W n ) = ωn  π∈N Ce (n)

c |V |  , V ∈π

2

or by the free cumulants, kn (W, ..., W ) = ωn c n2 , for all n ∈ N, where NCe (n) = {π ∈ NC (n) : |V | ∈ 2 N, ∀V ∈ π} in NC (n). Proof . Since j1 = ̸ j2 in Z, the semicircular elements Uj1 and Uj2 are distinct in the generating free semicircular family U of LQ , implying

108 ■ Constructive Analysis of Semicircular Elements

the freeness of them in LQ . So, the free random variable W is a free Poisson element of LQ by Eq. (1.3.14). So, by Eq. (3.5.1), the free Poisson distribution of W is characterized by the free moments,   |V |

τ (W n ) =

τ Uj2 π∈N C(n)

=

V ∈π

ω|V | c |V |  , π∈N Ce (n)

V ∈π

2

or by the free cumulants, ( ) kn (W, ..., W ) = τ Ujn2 = ωn c n2 , for all n ∈ N, by the semicircularity of Uj2 ∈ U in LQ . Similarly, one has the following result. Corollary 3.3: Suppose j1 ̸= j2 in Z, and let Uj1 ∈ U be a 2 semicircular element, and uj2 ∈ U, a ψ (qj2 ) -semicircular element of LQ . Then a free random variable, W = Uj1 uj2 Uj1 ∈ LQ is a free Poisson element whose free distribution is characterized by the free moments,   n τ (W n ) = ωn ψ (qj2 ) 

c |V |  , V ∈π

π∈N Ce (n)

2

or by the free cumulants,

(3.5.2) 2

kn (W, ..., W ) = ωn ψ (qj2 )

n 2

c n2 ,

for all n ∈ N. Proof . Since j1 ̸= j2 in Z, the and uj2 = ψ (qj2 ) Uj2 ∈ U are free weighted-semicircular element uj2 the free random variable W is a

free random variables Uj1 ∈ U , in LQ . Since ψ (qj2 ) ∈ R× , the is self-adjoint in LQ , and hence, well-defined free Poisson element

Semicircular Elements Induced by Orthogonal Projections ■ 109

by Eq. (1.3.14). So, the corresponding free Poisson distribution is characterized by the free moment,   τ (W n ) = ωn 

ψ (qj2 )

|V |

c |V |

V ∈π

π∈N Ce (n)

,

2

or by the free cumulants, ( ) n kn (W, ..., W ) = τ ujn2 = ωn ψ (qj2 ) c n2 , for all n ∈ N, by Eq. (3.5.1). Therefore, the free-distributional data (3.5.2) holds. Also, one can get the following corollary. Corollary 3.4: Suppose j1 = ̸ j2 in Z, and let uj1 , uj2 ∈ U be the corresponding weighted-semicircular elements of LQ . Then the free random variable, W = uj1 uj2 uj1 ∈ LQ is a free Poisson element, whose free distribution is characterized by the free moments,   2

τ (W n ) = ωn ψ (qj1 ) ψ (qj2 )

n

c |V |  ,

 π∈N Ce (n)

V ∈π

or by the free cumulants,

2

(3.5.3) 4

kn (W, ..., W ) = ωn ψ (qj1 ) ψ (qj2 )

2

n 2

c n2 ,

for all n ∈ N. Proof . Under hypothesis, two free random variables ujl = ψ (qjl ) Ujl , for l = 1 , 2 , are distinct in the free weighted-semicircular family U, and hence, they are free in LQ . Moreover, 2

W = Uj1 ψ (qj1 ) ψ (qj2 ) Uj2 Uj1 , 2

in LQ , where Uj1 and ψ (qj1 ) ψ (qj2 ) Uj2 are free in LQ , by the freeness of Uj1 , Uj2 ∈ U in LQ . Therefore, the free random variable W is a

110 ■ Constructive Analysis of Semicircular Elements 2

free Poisson element by Eq. (1.3.14) because ψ (qj1 ) ψ (qj2 ) Uj2 is selfadjoint in LQ (under Assumption 3.2). Thus, by Eq. (3.5.1), the free Poisson distribution of W is characterized by the free moments,   t|V | c |V |

τ (W n ) = ωn  π∈N Ce (n)

V ∈π

2

,

or by the free cumulants, ( )n n kn (W, ..., W ) = τ ((tUj2 ) ) = ωn t2 2 c n2 , for all n ∈ N, where 2

t = ψ (qj1 ) ψ (qj2 ) ∈ R× , in C. Therefore, the free-distributional data (3.5.3) holds true. The above corollaries characterize the free Poisson distributions on LQ induced by the family U ∪ U , where U is the free weightedsemicircular family (3.2.14), and U is the free semicircular family (3.2.18), generating LQ .

3.6

CERTAIN ∗-ISOMORPHISMS ACTING ON LQ

Even though this section is parallel to (or, motivated by) Section 2.10, the meanings and properties of the main results are different in the sense of operator-theoretic dynamical system theory (See Section 3.7 below). It demonstrates that even though our constructions of this section are motivated by and are similar to those of Section 2.10, the resulted objects and the corresponding structures are different from those of Chapter 2 mathematically. As before, let (A, ψ) be a C ∗ -probability space containing the family Q = {qj }j∈Z of mutually orthogonal projections generating the C ∗ -subalgebra Q = C ∗ (Q) of A, and let LQ be the semicircular filterization generated by the free semicircular family U of Eq. (3.2.18). Let h : Z → Z be a function defined by h (j ) = j + 1 , ∀j ∈ Z.

(3.6.1)

Semicircular Elements Induced by Orthogonal Projections ■ 111

Then it is easy to check the bijectivity of the function h. Indeed, it has its inverse, h−1 (j) = j − 1, ∀j ∈ Z. From the bijection h of Eq. (3.6.1), define a multiplicative linear transformation β on LQ by a linear morphism satisfying β (Uj ) = Uh(j) = Uj+1 , in U ,

(3.6.2)

in LQ . This morphism β of Eq. (3.6.2) is a well-defined bounded, bijective linear transformation because the restriction β |U to the generating free family U is bijective in LQ . Furthermore, by the multiplicativity of β, if N

Ujnl l ∈ LQ

W = l=1

is a free reduced word with its length-N in U , for an alternating sequence (j1 , ..., jN ) ∈ Z in the sense that: j1 ̸= j2 , j2 ̸= j3 , ..., jN −1 ̸= jN , in Z, and n1 , ..., nN ∈ N, one has that N

(3.6.3)

N ( ) N n β Ujnl l = (β (Ujl )) l = Ujnl l+1 ,

β (W ) = l=1

l=1

l=1

in LQ . By Eq. (3.6.3), it is realized that the freeness of W is preserved in LQ under the action of β, since the image β (W ) of an arbitrary free reduced word W is again a free reduced word with the same length-N in LQ by Eq. (3.2.21) because the N -tuple, (j1 + 1, ..., jN + 1) ∈ ZN is again an alternating sequence in Z. Proposition 3.4: The multiplicative linear transformation β of Eq. (3.6.2) is a ∗-isomorphism on LQ . Proof . Let β be the multiplicative linear transformation (3.6.2), satisfying (3.6.3). Observe that, if N

Ujnl l = Ujn11 Ujn22 ...UjnNN ∈ LQ

W = l=1

112 ■ Constructive Analysis of Semicircular Elements

is an arbitrary free reduced word with its length-N for N ∈ N, then the adjoint W ∗ of W is N n

W∗ =

UjNN−−l+1 = UjnNN ...Ujn22 Ujn11 , l+1 l=1

in LQ , by the self-adjointness of Uj1 , ..., UjN ∈ U , and hence, by Eq. (3.6.3), we have that ∗

N

β (W ) = l=1

n −l+1 UjNN−l+1 +1

N

= l=1

∗

Ujn1l+1

∗

= (β (W )) .

(3.6.4)

Recall that the semicircular filterization LQ is the free product Banach ∗-algebra (3.2.21). So, all elements of LQ are the limits of linear combinations of free reduced words in the generating free semicircular family U . Therefore, the operator-equality (3.6.4) implies that ∗

β (T ∗ ) = β (T ) , ∀T ∈ LQ . i.e., this bounded bijective multiplicative linear transformation β is a ∗-homomorphism on LQ , equivalently, it is a ∗-isomorphism on LQ . The above proposition shows that the morphism β of Eq. (3.6.2) is a ∗-isomorphism on LQ . Definition 3.7: Let (A1 , φ1 ) and (A2 , φ2 ) be topological ∗-probability spaces, and assume that η : A1 → A2 is a ∗-homomorphism. If η satisfies φ2 (η(a)) = φ1 (a), ∀a ∈ (A1 , φ1 ) , then this ∗-homomorphism η is said to be a free-homomorphism from (A1 , φ1 ) to (A2 , φ2 ). In such a case, we say (A1 , φ1 ) is freehomomorphic to (A2 , φ2 ) (via η) and denote this relation by (A1 , φ1 )

free-homo

−→

(A2 , φ2 ) .

If a free-homomorphism η is a ∗-isomorphism from A1 onto A2 , then it is called a free-isomorphism. In such a case, (A1 , φ1 ) is said to be free-isomorphic to (A2 , φ2 ) (via η), and this relation is denoted by (A1 , φ1 )

free-iso

=

(A2 , φ2 ) .

Semicircular Elements Induced by Orthogonal Projections ■ 113

By Definition 3.7, we have the following result. Theorem 3.9: A ∗-isomorphism β of Eq. (3.6.2) is a free-isomorphism on LQ . Proof . Let β be the ∗-isomorphism (3.6.2) on LQ by Proposition 3.4. Observe that, ( n ) ( ) n τ (β (Uj ) ) = τ Uj+1 = ωn c n2 = τ Ujn ,

(3.6.5)

for all n ∈ N, for all Uj ∈ U ⊂ LQ . It implies that N

τ

β l=1

Ujnl l

N

=τ l=1

Ujnl l+1 N

for all arbitrary free reduced words l=1

N

=τ l=1

Ujnl l ,

(3.6.6)

Ujnl l ∈ LQ with their length-N

in U , for all n1 , ..., nN , N ∈ N, by Eqs. (3.3.1), (3.3.2), (3.6.3) and (3.6.5). By Eq. (3.2.21), all free random variables of LQ are the limits of linear combinations of free reduced words in U . Therefore, by Eq. (3.6.6), τ (β (T )) = τ (T ) , ∀T ∈ LQ , and hence, this ∗-isomorphism β is a free-isomorphism on LQ . Like the free-isomorphism β of Eq. (3.6.2), define a multiplicative linear transformation β −1 on LQ by the linear morphism satisfying β −1 (Uj ) = Uh−1 (j) = Uj−1 , ∀j ∈ Z,

(3.6.7)

in U ⊂ LQ . Then, as above, it is a well-defined free-isomorphism on LQ . Corollary 3.5: A multiplicative linear transformation β −1 of Eq. (3.6.7) is a free-isomorphism on LQ . Proof . Similar to the proof of Proposition 3.4, this morphism β −1 of Eq. (3.6.7) is a well-defined ∗-isomorphism. Also, similar to the proof of Theorem 3.9, this ∗-isomorphism β −1 is a free-isomorphism on LQ .

114 ■ Constructive Analysis of Semicircular Elements

By Theorem 3.9 and Corollary 3.5, we have that β (LQ )

free-iso

=

free-iso

=

LQ

β −1 (LQ ).

(3.6.8)

{ } Define now a new ∗-isomorphisms β k k∈Z on LQ by

def

βk =

          

ββ...β

if k > 0

k-times

β −1 β −1 ...β −1

         

if k < 0

(3.6.9)

|k|-times

1LQ

if k = 0,

for all k ∈ Z, where 1LQ is the identity map on LQ , and (·) is the multiplication (or, the composition) of ∗-isomorphisms. Theorem 3.10: Every ∗-isomorphism β k of Eq. (3.6.9) is a freeisomorphism on LQ , for all k ∈ Z. i.e., β k (LQ )

free-iso

=

LQ , ∀k ∈ Z.

(3.6.10)

Proof . Let either k > 0 or k < 0 in Z. Then the free-isomorphic relation (3.6.10) holds by Eq. (3.6.8). If k = 0 , then it is not difficult to check that the identity map 1LQ , a well-defined ∗-isomorphism on LQ , is a free-isomorphism, since ( ) τ 1LQ (T ) = τ (T ) , ∀T ∈ LQ . So, the relation (3.6.10) holds for the case where k = 0 , too. Now, define a system, { } B = βk : k ∈ Z of the free-isomorphisms (3.6.9) in the automorphism group, Aut (LQ ) = {η : η is a ∗ -isomorphisms on LQ } ,

(3.6.11)

Semicircular Elements Induced by Orthogonal Projections ■ 115

of LQ under the multiplication of ∗-isomorphisms. Theorem 3.11: The subset B of Eq. (3.6.11) is a subgroup of the automorphism group Aut (LQ ). Moreover, the subgroup B is groupisomorphic to the infinite cyclic abelian group, i.e., Group

B = (Z, +) , in Aut (LQ ) ,

(3.6.12)

Group

where “ = ” means “being group-isomorphic to.” Proof . Let B be the subset (3.6.11) of Aut (LQ ). Then, by the very definition (3.6.9), β k1 β k2 = β k1 +k2 , in B, for all k1 , k2 ∈ Z, and hence, the multiplication (·) is closed on B. For any k1 , k2 , k3 ∈ Z, one has the associativity; ( k1 k2 ) k3 ( ) β β β = β k1 +k2 +k3 = β k1 β k2 β k3 , in B. Also, it contains the identity map 1LQ = β 0 in B, satisfying β k β 0 = β k+0 = β k = β 0 β k , for all k ∈ Z. So, the set B satisfies the inverse-property that: for each β k ∈ B, there exists a unique β −k ∈ B, such that β k β −k = β k+(−k) = β 0 = β −k+k = β −k β k , in B. Thus, the algebraic pair (B, ·) forms a subgroup of Aut (LQ ). Define now a function Φ : B → Z by ( ) Φ β k = k, ∀k ∈ Z. Then it is a well-defined bijection by Eq. (3.6.9) and (3.6.11). It satisfies that ( ) ( ) ( ) ( ) Φ β k1 β k2 = Φ β k1 +k2 = k1 + k2 = Φ β k1 + Φ β k2 , for all k1 , k2 ∈ Z. Therefore, this bijection Φ is a group-homomorphism, equivalently, it is a group-isomorphism, implying the isomorphic relation (3.6.12).

116 ■ Constructive Analysis of Semicircular Elements

{ } The above theorem shows that the family B = β k k∈Z of our freeisomorphisms on LQ not only forms a subgroup of the automorphism group Aut (LQ ) but also group-isomorphic to the infinite cyclic abelian group (Z, +). Since the free-isomorphisms of B act on LQ , and since B is a group, the family B provides a well-defined dynamical systemic structure on LQ . Definition 3.8: The free-isomorphisms β k of Eq. (3.6.9) are called the k -th integer shifts on the semicircular filterization LQ for all k ∈ Z. The group B of Eq. (3.6.11) of all integer shifts is called the integer-shift (sub)group (of Aut (LQ )). Remark 3.3: The abo{ ve Theorem 3.11 shows the differences between } the integer shifts B = β k k∈Z acting on the semicircular filterization LQ , and the pi-shifts S = {σk }k∈Z of Eq. (2.10.10) acting on the semicircular Adelic filterization LS. i.e., the pi-shift family S of Chapter 2 does not form an algebraic sub-structure in the homomorphism semigroup Hom (LS) of LS, the integer-shift group B forms a subgroup of the automorphism group Aut (LQ ) of LQ . Such differences allow us to study Section 3.7 below.

✥

3.7

GROUP-DYNAMICAL SYSTEM ON LQ

In Section { k } 3.6, we showed that there exists a well-defined group B = β k∈Z of the free-isomorphisms (3.6.9) acting on the semicircular filterization LQ , which is group-isomorphic to the infinite cyclic abelian group (Z, +) in the automorphism group Aut (LQ ). So, here, we study the corresponding group-dynamical system acting on LQ . 3.7.1

Discrete-Group Dynamical Systems

Let B be a topological ∗-algebra, and let λ be a group with its identity β0 . Assume that this group λ acts on B via a group-action α. i.e., for any β ∈ λ, the image α(β) becomes a ∗-isomorphism in the automorphism group Aut (B ), Aut (B ) = {g : B → B | g is a ∗ -isomorphism} .

Semicircular Elements Induced by Orthogonal Projections ■ 117

Definition 3.9: Suppose a topological ∗-algebra B , and a group λ are given as above. Then the mathematical triple, Γ

denote

=

(B, λ, α)

(3.7.1)

is called the group-dynamical system of λ acting on B via an action α. Let Γ be a group-dynamical system (3.7.1). Then one can define a crossed product topological ∗-algebra, BΓ

denote

=

B ⋊α λ,

(3.7.2)

by the Banach ∗-algebra generated by the Cartesian product set B × λ satisfying the α-relation: (a1 , β1 ) (a2 , β2 ) = (a1 α(β1 ) (a2 ) , β1 β2 ) and

(3.7.3) ( ) ∗ (a1 , β1 ) = α(β1 ) (a∗1 ) , β1−1 ,

for all a1 , a2 ∈ B , and β1 , β2 ∈ λ, where β1−1 is the group-inverse of β1 in λ. Suppose now a given topological ∗-algebra B induces a topological ∗-probability space (B, φ) for a bounded linear functional φ on B. Then, on the crossed product Banach ∗-algebra BΓ of Eq. (3.7.2), one can define a well-defined linear functional φΓ by the morphism satisfying that def

φΓ ((a, β)) = φ (α(β) (a))

(3.7.4)

for all a ∈ (B , φ), and β ∈ λ. Definition 3.10: Let Γ be a group-dynamical system (3.7.1), and BΓ , the corresponding crossed product Banach ∗-algebra (3.7.2) under the α-relation (3.7.3). Assume that there exists a well-defined linear functional φΓ of Eq. (3.7.4) on BΓ . Then the Banach ∗-probability space, BΓ

denote

=

(BΓ , φΓ ),

(3.7.5)

is called the Γ -dynamical (crossed product Banach-)∗-probability space.

118 ■ Constructive Analysis of Semicircular Elements

The Group-Dynamical System (B, LQ , α) { } In this section, let B = β k k∈Z be the integer-shift group on LQ , consisting of all integer shifts β k , for all k ∈ Z, which are freeisomorphisms on LQ by Eq. (3.6.12). By Eqs. (3.6.9) and (3.6.11), the integer-shift group B naturally act on the semicircular filterization LQ satisfying the free-isomorphic relation (3.6.12). So, one can define the corresponding group-dynamical system, 3.7.2

ΓQ = (B, LQ , α), where α is the group-action of B defined by ( ) α β k (T ) = β k (T ) , ∀T ∈ LQ ,

(3.7.6)

(3.7.7)

on the Banach ∗-algebra LQ , like (3.7.1). Thus, the corresponding crossed product Banach ∗-algebra, LQ

denote

=

LQ ⋊α B

(3.7.8)

is well-defined as a new Banach ∗-algebra with the corresponding αrelation, in the sense of Eq. (3.7.3) on LQ , satisfying that; ( )( ) ( ( ( ) ) ) T1 , β k1 T2 , β k2 = T1 α β k1 (T2 ) , β k1 β k2 , and

(3.7.9) ( )∗ ( ( ) ) T, β k = α β k (T ∗ ) , β −k ,

for all T1 , T2 , T ∈ LQ , and β k1 , β k2 , β k ∈ B. Theorem 3.12: Let LQ be the semicircular filterization, and B, the integer-shift group on LQ . Then the group-dynamical system ΓQ of Eq. (3.7.6) is well-defined, inducing the corresponding crossed product Banach ∗-algebra LQ of Eq. (3.7.8) with the α-relation (3.7.9). Also, this crossed product algebra LQ generates the Banach ∗-probability space, denote LQ = (LQ , τo ) , with the linear functional satisfying (3.7.10) ( ) ( ) τo T, β k = τ β k (T ) , ( ) for all T, β k ∈ LQ , where τ is the linear functional (3.2.15) on LQ .

Semicircular Elements Induced by Orthogonal Projections ■ 119

Proof . By Eqs. (3.7.4) and (3.7.5), to show the Banach ∗-algebra LQ is well-determined, it is sufficient to show that the linear morphism α of Eq. (3.7.7) is a well-defined group action of the integer-shift group B acting on LQ . But, by Proposition 3.4, (3.6.9), (3.6.11) and (3.6.12), indeed, α is a bounded multiplicative linear morphism from B into the automorphism group Aut (LQ ). Thus, the group-dynamical system ΓQ is well-defined, and hence, the Banach ∗-algebra LQ is well-defined. Therefore, since LQ = (LQ , τ ) is a well-defined Banach ∗-probability space (3.2.15), the Banach ∗-probability space LQ of Eq. (3.7.10) is well-established by Eqs. (3.7.4) and (3.7.5).

P

The above theorem shows that, whenever a fixed C ∗ -probability space (A, ψ) contains a family Q = {qj }j∈Z of mutually orthogonal projections, generating the C ∗ -subalgebra Q of A, having their free distributions {ψ (qj ) ∈ R× }j∈Z , one can construct not only the Banach ∗-probability space LQ , the semicircular filterization, generated by the free semicircular family U = {Uj }j∈Z induced by Q but also the Banach ∗-probability space, LQ = (LQ , τo )

of Eq. (3.7.10), induced by LQ and the integer-shift group B acting on LQ . Definition 3.11: Let LQ be the Banach ∗-probability space (3.7.10) induced by the group-dynamical system ΓQ of the integer-shift group B acting on the semicircular filterization LQ via the group-action α of Eq. (3.7.7). Then it is called the (integer-group-)dynamical semicircular (Banach-)∗-probability space (of B on LQ ). In the following section, we consider free-distributional data on this dynamical semicircular ∗-probability space LQ . To do that, we consider first a structure theorem of our dynamical ∗-probability space LQ . Let B be the integer-shift group. Since it is a discrete group (e.g., see (3.6.12)), one can construct the l 2 -Hilbert space,  H = l 2 (B) , generated by B \ β 0 ,

(3.7.11)

120 ■ Constructive Analysis of Semicircular Elements

{ } with its orthonormal basis B \ β 0 , equipped with the inner product ⟨, ⟩2 , < k1 k2 > β , β 2 = δk1 ,k2 , ∀k1 , k2 ∈ Z, inducing the l 2 -norm ∥.∥2 , satisfying ∥ξ∥2 =

⟨ξ, ξ⟩2 , ∀ξ ∈ H.

In the operator algebra B (H) of all bounded Hilbert-space operators on the Hilbert space H of Eq. (3.7.11), we define a C ∗ -subalgebra, B = C ∗ (B) of B (H),

(3.7.12)

generated by B, i.e., B is a group C ∗ -algebra of the integer-shift group B in B (H). Then, this group C ∗ -algebra B of Eq. (3.7.12) canonically acts on B (H). i.e., every operator T of B is expressed by T =

tk β k , with tk = tβ k ∈ C, β k ∈B

with its adjoint, T∗ =

tk β −k ∈ B, β k ∈B

where H by

is the infinite (or, the limit of finite) sum(s), and it acts on ( ) tk β k ξ , in H.

T (ξ) = β k ∈B

i.e., the integer shifts β k ∈ B are acting on the Hilbert space H of Eq. (3.7.11) as unitaries with their adjoints ( k )∗ β = β −k , ∀k ∈ Z. Recall that a unitary (or a unitary operator) u on an arbitrary Hilbert space H is an operator satisfying u∗ = u−1 , or u∗ u = uu∗ = IH on H , where IH is the identity operator on H . Then, one can construct the tensor product Banach ∗-algebra, LQ ⊗ B,

Semicircular Elements Induced by Orthogonal Projections ■ 121

where ⊗ is the tensor product of Banach ∗-algebras, and define the Banach ∗-subalgebra, L

denote

=

LQ ⊗α B of LQ ⊗ B,

(3.7.13)

by the “conditional” tensor product of LQ and B , satisfying the α-condition:




T1 ⊗ β k1 T2 ⊗ β k2 = T1 α β k1 (T2 ) ⊗ β k1 β k2 , and

(3.7.14) T ⊗ βk


∗



= α β k (T ∗ ) ⊗ β −k ,

for all T1 , T2 , T ∈ LQ , and β k1 , β k2 , β k ∈ B ⊂ B. By Eqs. (3.7.13) and (3.7.14), indeed, the Banach ∗-algebra L is a well-defined ∗-subalgebra of LQ ⊗ B. Lemma 3.4: The crossed product Banach ∗-algebra LQ of Eq. (3.7.8), inducing the dynamical semicircular ∗-probability space (3.7.10), and the conditional tensor product Banach ∗-algebra L of Eq. (3.7.13) are ∗-isomorphic. i.e., ∗-iso

LQ = L .

(3.7.15)

Proof . Define a linear transformation Ψ : LQ → L by the linear morphism satisfying

Ψ Uj , β k = Uj ⊗ β k , for all j , k ∈ Z. Then, by the very definition, it is a bounded, bijective linear transformation, since the restriction from the generating set
 Uj , β k : j, k ∈ Z of LQ to the generating set 

Uj ⊗ β k : j, k ∈ Z of L

is a bijection. Also, this morphism satisfies that, for all j1 , j2 , k1 , k2 ∈ Z,



Ψ Uj1 , β k1 Uj2 , β k2 = Ψ Uj1 β k1 (Uj2 ) , β k1 +k2

122 ■ Constructive Analysis of Semicircular Elements

by the α-relation (3.7.9) = Uj1 Uj2 +k1 ⊗ β k1 +k2 ( )( ) = Uj1 ⊗ β k1 Uj2 ⊗ β k2 by the α-condition (3.7.14) =Ψ

(( )) (( )) Uj1 , β k1 Ψ Uj2 , β k2 ,

implying that Ψ (T1 T2 ) = Ψ (T1 ) Ψ (T2 ) in L , for all T1 , T2 ∈ LQ . Also, one has that, for all t ∈ C, and j , k ∈ Z, Ψ

( ( ) ) ( ( ))∗ t Uj , β k = tΨ β k Uj∗ , β −k

by Eq. (3.7.9) ( ) ( ) = tΨ Uj+k , β −k = t Uj+k ⊗ β −k ( )∗ ( ( ))∗ = t Uj ⊗ β k = Ψ t Uj , β k , by Eq. (3.7.14), implying that ∗

Ψ (T ∗ ) = Ψ (T ) , in L , for all T ∈ LQ . Thus, the bounded bijective linear transformation Ψ is a ∗-homomorphism from LQ onto L , equivalently, it is a ∗-isomorphism. Therefore, the isomorphic relation (3.7.15) holds. Let L = LQ ⊗α B be the conditional tensor product Banach ∗algebra (3.7.13), ∗-isomorphic to LQ . Define a linear functional τ o on L by the linear morphism satisfying τ o (T ⊗ β) = τ (β (T )) , ∀T ⊗ β ∈ L ,

(3.7.16)

where τ is the linear functional on the semicircular filterization LQ .

Semicircular Elements Induced by Orthogonal Projections ■ 123

Such a linear functional τ o of Eq. (3.7.16) is well-defined on L by Eq. (3.7.13), and hence, the corresponding Banach ∗-probability space, denote

L

=

(L , τ o ) ,

(3.7.17)

is well-defined. Observe now that, for generating operators, ( ) Ujl , β kl ∈ LQ , for l = 1 , ..., n, for n ∈ N, n

(

) ( ) Ujl , β kl = Uj1 Uj2 +k1 Uj3 +k1 +k2 ...Ujn +k1 +...+kn−1 , β k1 +...+kn ,

l=1

in LQ by the induction on (3.7.9), i.e.  n n ( ) kl  Ujl , β = U l=1

l=1



n

kl

l−1

jl +

ki

, β l=1  ,

i=1

in LQ , with axiomatization:

(3.7.18) 0

ki = 0. i=1

Similarly, for generating operators, Ujl ⊗ β kl ∈ L , for l = 1 , ..., n, for n ∈ N, one has  n

Ujl ⊗ β l=1

kl



n

= l=1

U

l−1

jl +

ki

⊗

n

β l=1

kl

,

(3.7.19)

i=1

in L , by the induction on (3.7.14), with the same axiomatization with (3.7.18). Theorem 3.13: The dynamical semicircular ∗-probability space LQ = (LQ , τo ) of Eq. (3.7.10) is free-isomorphic to the Banach ∗-probability space L = (L , τ o ) of Eq. (3.7.17). i.e.,

124 ■ Constructive Analysis of Semicircular Elements

LQ

free-iso

=

L.

(3.7.20)

Proof . By Lemma 3.4, the Banach ∗-algebras LQ and L are ∗-isomorphic via the ∗-isomorphism Ψ, intro(duced in ) the proof. Consider that, for any generating operators Ujl , β kl ∈ LQ , for l = 1 , ..., n, for n ∈ N (where either j1 , ..., jn , or k1 , ..., kn are not necessarily distinct from each other in Z, if n > 1 ), we have that    n n ( n kl ) τo Ψ Ujl , β kl = τ o Ψ  U l−1 , β l=1  l=1

jl +

l=1

ki

i=1

 o



n

= τ 

U l=1

l−1

jl +

ki



n

⊗

β

kl l=1



i=1

by the definition of Ψ 



n

= τ  β l=1

kl



n

U

 l=1

l −1

jl +

ki



i=1

by Eq. (3.7.16)  = τo  l=1



n

n

kl

U

l −1

jl +

ki

, β l=1 ,

(3.7.21)

i=1

by Eq. (3.7.10). Thus, by Eqs. (3.5.1) and (3.7.21), the ∗-isomorphism Ψ preserves the free-distributional data, i.e., it is a free-isomorphism. Therefore, the free-isomorphic relation (3.7.20) holds. The above theorem shows that not only two Banach ∗-algebras LQ and L are ∗-isomorphic but also they are free-isomorphic, i.e., they are regarded as the same free-probabilistic structures. Notation 3.1: By the free-isomorphic relation (3.7.20), from below, we use two Banach ∗-probability spaces LQ and L as an identical Banach ∗-probability space, called the dynamical semicircular ∗-probability space, and denote both by LQ . Sometimes, we use crossed product terminology, and sometimes, conditional tensor product terminology case-by-case alternatively, if there are no confusions.

✥

Semicircular Elements Induced by Orthogonal Projections ■ 125

3.7.3

The Dynamical Semicircular ∗-Probability Space LQ

In this section, we study free-distributional data on the dynamical semicircular ∗-probability space, LQ = (LQ ⋊α B, τo ) ,

induced by the group-dynamical system, ΓQ = (B, LQ , α) , where B is the integer-shift group acting on the semicircular filterization LQ via the group-action α of (3.7.7). Recall that   n n ( n kl ) Ujl , β kl =  U l−1 , β l=1  , (3.7.22) l=1

l=1

jl +

ki

i=1

( ) in LQ for the generating free random variables Ujl , β kl , for l = 1 , ..., n, for all n ∈ N, by Eq. (3.7.18) or by Eq. (3.7.19). ( ) Proposition 3.5: Let Tl = Ujl , β kl ∈ LQ be generating free random variables for l = 1 , ..., n, for n ∈ N, where either j1 , ..., jn , or k1 , ..., kn n

are not necessarily mutually distinct in Z. If W =

Tl ∈ LQ , then

l=1





n

τo (W ) = τ 

U

l−1

jl +

l=1

ki

,

(3.7.23)

i=1

where the right-hand side of Eq. (3.7.23) is characterized by Eq. (3.5.1) or by Eq. (1.4.13). Proof . For convenience, let k0

n

denote

=

kl in Z. Then l=1





n

τo (W ) = τo  l=1

U

l−1

jl +

ki

, β k0 

i=1

by Eq. (3.7.22) 



= τ β k0 



n

U l=1

l −1

jl +

ki i=1

by Eq. (3.7.10) or (3.7.16)



126 ■ Constructive Analysis of Semicircular Elements





n

= τ 

U

l=1

l −1

jl +

ki

,

(3.7.24)

i=1

by Eq. (3.6.10) (i.e., β k0 is a free-isomorphism on LQ ). Moreover, this quantity (3.7.24) is characterized by Eq. (3.5.1) (or by Eq. (1.4.13)) under the universality of the semicircular law. Therefore, the free-distributional data (3.7.23) holds by Eq. (3.7.24). The above proposition provides the general joint free moment computation (3.7.23). We study some special cases of Eq. (3.7.23) in the rest of this section. ( ) Theorem 3.14: Let U = Uj , β 0 ∈ LQ be a generating free random variable, for j ∈ Z, where β 0 = 1LQ is the identity of B. Then U is semicircular in LQ . i.e., (

) Uj , β 0 are semicircular in LQ , ∀j ∈ Z.

(3.7.25)

Proof . A generating free random variable U is self-adjoint in the dynamical semicircular ∗-probability space LQ , since ( ( ) ) ( ) ( ) U ∗ = β 0 Uj∗ , β −0 = Uj−0 , β 0 = Uj , β 0 = U, in LQ . Consider that 

0 + 0 + ... + 0

 U n = Uj Uj+0 Uj+0+0 ...Uj+0 + 0 + ... + 0 , β

n-times

  ( n 0)  = Uj , β ,

(n−1)-times

in LQ , by Eq. (3.7.22). So, one can have that ( ( )) ( ) τo (U n ) = τ β 0 Ujn = τ Ujn = ωn c n2 , for all n ∈ N. Therefore, this self-adjoint free random variable U is semicircular in LQ . ( ) It is not hard to check a generating free random variable Uj , β k is in LQ in general. In particular, if k = ̸ 0 in Z, then ( not ksemicircular ) Uj , β cannot be semicircular because

Semicircular Elements Induced by Orthogonal Projections ■ 127

(

Uj , β k

)∗

( ( ) ) ( ) ( ) = β k Uj∗ , β −k = Uj+k , β −k = ̸ Uj , β k , (3.7.26)

in LQ . By the y (3.7.25) of the generating free random {( semicircularit )} variables Uj , β 0 j∈Z , we are now interested in “non-semicircular” generating free random variables of LQ , satisfying (3.7.26). Suppose k= ̸ 0 , and let ( ) Wks = Ujs , β k ∈ LQ , for k ̸= 0, and s ∈ N. (3.7.27) If Wks is a generating operator (3.7.27) of LQ , then 

k + k + ... + k

 n s s s Uj+k+k ...Uj+ (Wks ) = Ujs Uj+k k + k + ... + k , β

n-times

  ,

(n−1)-times

i.e.,

(3.7.28) (

Ujs , β k

)n

n

Ujs+(l−1 )k , β nk

=

,

l=1

in LQ , for all n ∈ N. Also, by Eqs. (3.7.26) and (3.7.28), we have (

∗ )n

(Wks )

( s )n s s s = Uj+k , β −k = Uj+k Ujs Uj−k ...Uj−(n−2)k , β −nk ,

i.e.,

(3.7.29) (

)∗ n Ujs , β k

n s Uj−(l−2)k , β −nk

=

,

l=1

in LQ , for all n ∈ N. ( ) Theorem 3.15: Suppose k ̸= 0 , and let Wks = Ujs , β k be a generating free random variable (3.7.27) of the dynamical semicircular ∗-probability space LQ , for j ∈ Z. Then (( )n ) ( )n ( s k )∗ n τo Ujs , β k = ωs c 2s = τo Uj , β , (3.7.30) for all n ∈ N.

128 ■ Constructive Analysis of Semicircular Elements

Proof . For n ∈ N, if Wks ∈ LQ is a generating operator (3.2.27), then n

n

τo ((Wks ) ) = τo

l=1

s Uj+(l−1)k , β nk

by Eq. (3.7.28) =τ

β nk

n l=1

n

Ujs+(l−1 )k

=τ l=1

Ujs+(l−1 )k

by Eq. (3.6.10) ( )n = τ Ujs Ujs+k Ujs+2k ...Ujs+(n−1 )k = ωs c 2s ,

(3.7.31)

by Eq. (3.5.1) (or by Eq. (1.4.13)) because the index n-tuple, J = (j, j + k, j + 2k, ..., j + (n − 1)k) induces the corresponding noncrossing partition, πJ = {(j), (j + k), (j + 2k), ..., (j + (n − 1)k)} , in the noncrossing-partition lattice NC ({j , j + k , ..., j + (n − 1 )k )}), since k ̸= 0 . i.e., n

( )n s τ Uj+(l−1)k = ωs c 2s ,

s s ...Uj+(n−1)k = τ Ujs Uj+k l=1

by the semicircularity of Uj , Uj +k ..., Uj +(n−1 )k ∈ U in LQ . Similar to Eq. (3.7.31), we have τo

((

(Wks )

∗ )n )

= τo

(

s Uj+k , β −k

)n

( )n = ωs c 2s ,

(3.7.32)

for all n ∈ N. Or, independent from (3.7.32), one can get that ( )n n τ ((Wk∗ ) ) = τo (Wkn ) = ωs c 2s , for all n ∈ N, since ωn , c 2s ∈ R for all s ∈ N. Therefore, the free-distributional data (3.7.30) holds by Eqs. (3.7.31) and (3.7.32).

Semicircular Elements Induced by Orthogonal Projections ■ 129

Now, consider that, if Wks ∈ LQ is in the sense of Eq. (3.2.27), then ( )( ) ( ) ∗ Wks (Wks ) = Ujs , β k Ujs+k , β −k = Ujs Ujs+2k , β 0 , and

(3.7.33) ( s )( ) ( s ) ∗ s (Wks ) Wks = Uj+k , β −k Ujs , β k = Uj+k Uj−k , β0 ,

by Eq. (3.7.9) (or, by Eq. (3.7.14)). Lemma 3.5: Let Wks be a generating free random variable (3.2.27) of LQ , for k ∈ Z \ {0 } and s ∈ N. Then n denote

(Yks )

=

(

Wks (Wks )

∗ )n

=

((

s Ujs Uj+2k

)n

) , β0 ,

and

(3.7.34) (Zks )

n denote

=

( s ∗ s )n (( s )n 0 ) s (Wk ) Wk = Uj+k Uj−k ,β ,

in LQ , for all n ∈ N. Proof . Suppose Yks = Wks Wks ∗ , and Zks = Wks ∗ Wks are the free random variables (3.7.34) of the dynamical semicircular ∗-probability ̸ 0 in Z. Then, for any n ∈ N, space LQ , for k =   0 + ... + 0 0 + ... + 0 ( ) ( ) ( ) n s 0 s s n−1 , n (Yks ) =  Ujs Uj+2 Ujs Uj+2k ...β Ujs Uj+2k ,β k β

and   0 + ... + 0 0 + ... + 0 ( ) ( ) ( ) s s s s s s n−1 , n (Zks )n =  Uj+k Uj−k β 0 Uj+k Uj−k ...β Uj+k Uj−k ,β

by Eqs. (3.7.9) and (3.7.33), implying the operator-equality (3.7.34) in LQ . By Eq. (3.7.34), we obtain the following result. ( ) Proposition 3.6: Let Wks = Ujs , β k be a generating operator ̸ 0 , and let Yks = Wks Wks ∗ and Zks = Wks ∗ Wks be the (3.7.27) for k = free random variables (3.7.34) in LQ . Then

130 ■ Constructive Analysis of Semicircular Elements

( )( )n n n s ω c = τo ((Zks ) ) , τo ((Yks ) ) = ωns c ns s 2 2

(3.7.35)

for all n ∈ N. Proof . Under hypothesis, we have n

(Yks ) =

((

s Ujs Uj+2k

)n

) , β0 ,

and n

(Zks ) =

((

s Ujs+k Uj−k

)n

) , β0 ,

in LQ , for all n ∈ N. So, n

τo ((Yks ) ) = τ

((

s Ujs Uj+2k

)n )

,

and

(3.7.36) n

τo ((Zks ) ) = τ

((

s s Uj+k Uj−k

)n )

,

for all n ∈ N, characterized by Eq. (3.5.1). By Eq. (3.7.36), observe that τ

((

s Ujs Uj+2k

)n )

( ) s = τ Ujs Ujs+2k Ujs Ujs+2k ...Ujs Uj+2k (( )n ) ( ( s ))n ( ( ns )) n = τ Ujs τ Uj+2k = τ Uj (τ (Uj+2k )) , (3.7.37)

by Eq. (3.5.1) (or by Eq. (1.4.13)) because the free random variable s Ujs Uj+2k

n

∈ LQ induces the (2ns)-tuple, J = (Jj , Jj+2k , Jj , Jj+2k , ..., Jj Jj+2k ) ,

with





Jj = j, j, ..., j  , s-times

and





Jj+2k = j + 2k, ..., j + 2k  , s-times

Semicircular Elements Induced by Orthogonal Projections ■ 131

in J , and hence, this (2ns)-tuple J generates the corresponding noncrossing partition,      πJ = Jj , Jj , ..., Jj  , (Jj+2k ) , ..., (Jj+2k ) ,   n-times

n-times

in NC (2ns), satisfying (( )n ) ( ( ns )) ( ( s ))n s τ Ujs Uj+2k = τ Uj τ Uj+2k , for n ∈ N. Similar to Eq. (3.7.37), one can get that ( ( ns )) n n τo ((Zks ) ) = τ Uj+k (τ (Uj−k )) ,

(3.7.38)

for all n ∈ N. Therefore, by Eqs. (3.7.37), (3.7.38), and the semicirculariry, the free-distributional data (3.7.35) are obtained. More generally, the following result is obtained. ( ) Theorem 3.16: Let Wksl = Ujsl , β k be a generating operator (3.7.27) for k ∈ Z \ {0 } and sl ∈ N, for l = 1 , 2 , and let Yk = Wks1 Wks2 ∗ and Zk = Wks2 ∗ Wks1 be the free random variables (3.7.34) in LQ . Then τo (Ykn ) = ωns1 c ns21

n

ωs2 c s22

,

and

(3.7.39) τo (Zkn ) = ωns2 c ns22

n

ωs1 c s21

,

for all n ∈ N. Proof . Under hypothesis, we have that ( ) s2 s2 Yk = Ujs1 , β k Uj+k , β −k = Ujs1 Uj+2k , β0 , and s2 Zk = Uj+k , β −k

(

) s2 s1 Ujs1 , β k = Uj+k Uj−k , β0 ,

132 ■ Constructive Analysis of Semicircular Elements

and hence, Ykn =

s2 Ujs1 Uj+2k

n

, β0 ,

and Zkn =

s2 s1 Uj+k Uj−k , β0 ,

in LQ , for all n ∈ N. Thus, one has that τo (Ykn ) = τ

s2 Ujs1 Uj+2k

τo (Zkn ) = τ

s2 s1 Uj+k Uj−k

n

n

= ωns1 c ns21

ωs2 c s22

= ωns2 c ns22

ωs1 c s21

,

and n

n

,

for all n ∈ N, similar to the proof of Eq. (3.7.35). Therefore, the free-distributional data (3.7.39) holds. The above free-distributional data (3.7.30), (3.7.35) and (3.7.39) provide the building blocks of computing the joint free distribution of the generating free random variables Wks ∈ LQ of Eq. (3.7.27), as special cases of Eq. (3.7.23).

3.8

BANACH ∗-PROBABILITY SPACES LQ [N ]

Let LQ be the semicircular filterization generated by the free semicircular family, U =

Uj =

1 (l ⊗ qj ) ∈ LQ (j) : j ∈ Z , ψ (qj )

induced by the family Q = {qj }j∈Z of mutually orthogonal projections in a C ∗ -probability space (A, ψ), whose free distributions are { } ψ (qj ) ∈ R× : j ∈ Z . Now, fix an arbitrary natural number N ∈ N, and take the free semicircular (sub-)family UN (of U ) by UN = {U1 , ..., UN } ,

(3.8.1)

in LQ . By the mutually distinctness of U1 , ..., UN ∈ UN in U , these semicircular elements are mutually free from each other in LQ .

Semicircular Elements Induced by Orthogonal Projections ■ 133

Thus, by the structure theorem (3.2.21) of LQ , one can define the Banach ∗-subalgebra LQ [N ] by def N

LQ [N ] = ⋆ C [{Ul }], l=1

(3.8.2)

in LQ . i.e., this Banach ∗-subalgebra LQ [N ] of Eq. (3.8.2) is a ∗-subalgebra of LQ generated by the free semicircular sub-family UN of (3.8.1) over C = C · 1LQ , containing its unity, 1N

denote

=

1LQ [N ] = 1LQ .

Definition 3.12: The Banach ∗-subalgebra LQ [N ] of Eq. (3.8.2) is called the semicircular N -(sub-)filterization (of LQ ). On this sub-structure LQ [N ], one can construct the free-probabilistic substructure, ) denote ( LQ [N ] = LQ [N ], τN = τ |LQ [N ] , (3.8.3) of LQ . This sub-structure LQ [N ] of Eq. (3.8.3) is also called the semicircular N -filterization (generated by UN of Eq. (3.8.1)). By the definitions (3.8.2) and (3.8.3) of the semicircular N -filterizations {LQ [N ]}N ∈N , we have the following theorem. Theorem 3.17: Let (B, φ) be a unital Banach ∗-probability space N generated by the free semicircular family S = {xj }j=1 , containing its unity 1B ∈ B , for N ∈ N. If φ (1B ) = τN (1N ) , then

(3.8.4) (B, φ)

free-iso

=

LQ [N ].

Proof . Suppose a Banach ∗-algebra B is generated by the free N semicircular family S = {xj }j=1 . Then, by the freeness on the generator set S, N

B = ⋆ C [{xj }], j=1

134 ■ Constructive Analysis of Semicircular Elements

the free product Banach ∗-algebra with its free blocks C [{xj }], where Y are the Banach-topology closures of subsets Y of B , by the selfadjointness of xj ∈ S in B . So, one can define a bounded multiplicative linear transformation, Φ : B → LQ [N ], by a morphism satisfying Φ (xj ) = Uj , ∀j = 1, ..., N, N

where UN = {Uj }j=1 is the generating free semicircular family of the semicircular N -filterization LQ [N ] of Eq. (3.8.2). Then, it is a welldefined ∗-isomorphism, i.e., two Banach ∗-algebras B and LQ [N ] are ∗-isomorphic from each other. Now, let n

xnj1l ∈ (B, φ)

W = l=1

be a free reduced words with its length-n. Then n

Ujnl l ∈ LQ [N ]

Φ (W ) = l=1

is a free reduced words with its length-n, too, by the multiplicativity of the ∗-isomorphism Φ. Consider that τ (Φ (W )) = φ (W ) , by Eq. (1.4.13). Since W ∈ (B , φ) is arbitrary and since φ (1B ) = τ (1N ) , (by regarding 1B as a free reduced word with its length-0), and since all free random variables of (B , φ) are the limits of linear combinations of free reduced words in S, the above computation guarantees that τ (Φ (T )) = φ (T ) , ∀T ∈ (B, φ) . Therefore, two Banach ∗-probability spaces (B, φ) and LQ [N ] are freeisomorphic, equivalently, the relation (3.8.4) holds true.

Semicircular Elements Induced by Orthogonal Projections ■ 135

Remark 3.4: By the free-isomorphic relation (3.8.3), the Banach ∗probability space (B, φ) and the semicircular N -filterization LQ [N ] are understood as the identical Banach ∗-probability spaces, whenever their unities have the same free distribution. i.e., our semicircular N -filterization LQ [N ] becomes a free-probabilistic representative, or a candidate of all Banach ∗-probability spaces generated by N many mutually free semicircular elements having the identical free distributions of their unities. By Remark 3.4, from below, we consider the semicircular N filterization LQ [N ] as a representative of all Banach ∗-probability spaces generated by N -many mutually free semicircular elements having the free distributions of their unities, identical to τ (1N ). i.e., we regard LQ [N ] not only as a free-probabilistic sub-structure of the semicircular filterization LQ but also as an independent freeprobabilistic structure. In the rest of this section, we focus on free-isomorphic relations among {LQ [N ]}N ∈N . Theorem 3.18: If N1 ≤ N2 in N, then LQ [N1 ]

free-homo

−→

LQ [N2 ].

(3.8.5)

Proof . Suppose N1 ≤ N2 in N, and let N

l UNl = {Uj }j=1

be the corresponding free semicircular families, generating the semicircular Nl -filterizations LQ [Nl ], for l = 1 , 2 . Define a multiplicative linear transformation, : LQ [N1 ] → LQ [N2 ] by a morphism satisfying (Uj ) = Uj , ∀j = 1, ..., N1 . Then it is a well-defined embedding map from LQ [N1 ] into LQ [N2 ], and hence, it is a well-defined injective ∗-homomorphism. Furthermore, by the very definition, if W ∈ LQ [N1 ] is a free reduced word in UN1 , then τN2

(W ) = τN2 (W ) = τN1 (W ) ,

by Eqs. (3.8.2) and (3.8.3). It implies that

136 ■ Constructive Analysis of Semicircular Elements

(T ) = τN1 (T ) , ∀T ∈ LQ [N1 ],

τN2

in LQ [N2 ]. Therefore, the free-homomorphic relation (3.8.5) holds. By Eq. (3.8.5), one immediately obtain the following freehomomorphic chain. Corollary 3.6: If N1 ≤ N2 ≤ ... ≤ Nn in N, for n ∈ N>1 = N \ {1 }, then LQ [N1 ]

free-homo

−→

LQ [N2 ]

free-homo

−→

···

free-homo

−→

LQ [Nn ].

(3.8.6)

Proof . The proof of the free-homomorphic chain (3.8.6) is done by the induction on (3.8.5). By Eqs. (3.8.5) and (3.8.6), whenever N1 ≤ N2 in N, LQ [N1 ]

free-homo

−→

LQ [N2 ].

Then, how about the converse? Remark 3.5: We do not know the converse of (3.8.5) holds, or not, yet, because of the same difficulties with the main result of Ref. [19]: The Radulescu’s alternative theorem of [19] says that if L (Fn ) are the free group factors (the group von Neumann algebras) of the free groups Fn of n-generators for n ∈ N∞ >1 = (N \ {1 }) ∪ {∞} , equipped with their canonical traces trn (as W ∗ -probability spaces), then either ∗-iso L (Fn ) = L (F∞ ) , ∀n ∈ N∞ >1 , or

(3.8.7) ∗-iso

L (Fn1 ) = ̸ L (Fn2 ) , ∀n1 = ̸ n2 ∈ N∞ >1 . holds true. In Eq. (3.8.7), the ∗-isomorphic relation actually implies the free-isomorphic relation automatically because L (Fn ) are factors! We still do not know ∗-iso

L (F2 ) = L (F3 ) ,

Semicircular Elements Induced by Orthogonal Projections ■ 137

equivalently, we do not know

(3.8.8) ∗-homo

L (F2 ) ←− L (F3 ) , even though ∗-homo

L (F2 ) −→ L (F3 ) holds true. Since they are factors, the above relation (3.8.8) implies that free-iso L (F2 ) = L (F3 ) , respectively, L (F2 )

free-homo

←−

L (F3 ) .

If one can find which one of Eq. (3.8.7) holds (from the consideration on (3.8.8)), then our question will be similarly answered. Or, if one can answer our question, then we similarly are able to show which statement of Eq. (3.8.7) is true.

REFERENCES [1] D. Alpay, P. E. T. Jorgensen, and G. Salomon, On Free Stochastic Processes and Their Derivatives, Stochastic Processes Appl., 124, no. 10, (2014) 3392–3411. [2] D. Alpay, P. E. T. Jorgensen, and D. Levanony, On the Equivalence of Probability Spaces, J. Theoret. Probab., 30, no. 3, (2017) 813–841. [3] D. Alpay, and P. E. T. Jorgensen, Spectral Theory for Gaussian Processes: Reproducing Kernels, Boundaries, and L2-Wavelet Generators with Fractional Scales, Numer. Funct. Anal. Optim., 36, no. 10, (2015) 1239–1285. [4] D. Alpay, P. E. T. Jorgensen, and D. Levanony, A Class of Gaussian Processes with Fractional Spectral Measures, J. Funct. Anal., 261, no. 2, (2011) 507–541. [5] M. Ahsanullah, Some Inferences on Semicircular Distribution, J. Stat. Theory Appl., 15, no. 3, (2016) 207–213. [6] H. Bercovici, and D. Voiculescu, Superconvergence to the Central Limit and Failure of the Cramer Theorem for Free Random Variables, Probab. Theory Related Fields, 103, no. 2, (1995) 215–222.

138 ■ Constructive Analysis of Semicircular Elements [7] M. Bozejko, W. Ejsmont, and T. Hasebe, Noncommutative Probability of Type D, Int. J. Math., 28, no. 2, (2017) 30. [8] M. Bozheuiko, E. V. Litvinov, and I. V. Rodionova, An Extended Anyon Fock Space and Non-commutative Meixner-Type Orthogonal Polynomials in the Infinite-Dimensional Case, Uspekhi Math. Nauk., 70, no. 5, (2015) 75–120. [9] I. Cho, Free Semicircular Families in Free Product Banach *-Algebras Induced by p-Adic Number Fields over Primes p, Compl. Anal. Oper. Theory, 11, no. 3, (2017) 507–565. [10] I. Cho, Semicircular-Like Laws and the Semicircular Law Induced by Orthogonal Projections, Compl. Anal. Oper. Theory, 12, (2018) 1657– 1695. [11] I. Cho, and P. E. T. Jorgensen, Semicircular Elements Induced by Projections on Separable Hilbert Spaces. In: D. Alpay, and M. Vajiac (eds.), Operator Theory Advances & Applications. Birkhauser: Cham (2018). DOI: 10.1007/978-3-030-18484-1-6. [12] I. Cho, and P. E. T. Jorgensen, Semicircular Elements Induced by p-Adic Number Fields, Opuscula Math., 35, no. 5, (2017) 665–703. [13] A. Connes, Noncommutative Geometry. Academic Press: San Diego, CA (1994). ISBN: 0-12-185860-X, [14] T. Gillespie, Prime Number Theorems for Rankin-Selberg L-Functions over Number Fields, Sci. China Math., 54, no. 1, (2011) 35–46. [15] P. R. Halmos, Graduate Texts in Mathematics: Hilbert Space Problem Books. Springer (1982). ISBN: 978-0387906850. [16] B. Meng, and M. Guo, Operator-Valued Semicircular Distribution and its Asymptotically Free Matrix Models, J. Math. Res. Exposition, 28, no. 4, (2008) 759–768. [17] I. Nourdin, G. Peccati, and R. Speicher, Multi-Dimensional Semicircular Limits on the Free Wigner Chaos, Progr. Probab., 67, (2013) 211–221. [18] V. Pata, The Central Limit Theorem for Free Additive Convolution, J. Funct. Anal., 140, no. 2, (1996) 359–380. [19] F. Radulescu, Random Matrices, Amalgamated Free Products and Subfactors of the C*-Algebra of a Free Group of Nonsingular Index, Invent. Math., 115, (1994) 347–389.

Semicircular Elements Induced by Orthogonal Projections ■ 139 [20] P. Shor, Quantum Information Theory: Results and Open Problems, Geom. Funct. Anal (GAFA), Special Volume: GAFA2000, (2000) 816– 838. [21] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Am. Math. Soc. Mem., 132, no. 627, (1998) x+88. [22] R. Speicher, Free Probability and Random Matrices, Proceedings of the International Congress of Mathematicians, Seoul, vol. III, Kyung Moon Sa, (2014) 477–501. [23] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Series on Soviet & East European Mathematics. World Scientific: Singapore (1994). ISBN: 978-981-02-0880-6. [24] D. Voiculescu, Free Probability for Pairs of Faces I, Comm. Math. Phy., 332, no. 3, (2014) 955–980. [25] D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, vol. 1. American Mathematical Society: Ann Arbor, MI (1992). ISBN-13: 978-0821811405. [26] Y. Yin, Z. Bai, and J. Hu, On the Semicircular Law of Large-Dimensional Random Quaternion Matrices, J. Theo. Probab., 29, no. 3, (2016) 1100– 1120. [27] Y. Yin, and J. Hu, On the Limit of the Spectral Distribution of LargeDimensional Random Quaternion Covariance Matrices, Random Mat. Theo. Appl., 6, no. 2, (2017) 20.

CHAPTER

4

Certain Banach-Space Operators on LQ [N]

or more about basic free-probabilistic tools and techniques, see e.g., [15–25]. Throughout this section, we use the same notations and terminology of Chapter 3 (under the same assumptions). Motivated by Sections 2.10 and 2.11 (e.g., also, see Refs. [2,6]), we study certain Banach-space operators acting on our Banach ∗-probability spaces,

F

{LQ } ∪ {LQ [N ]}N ∈N , generated by the free semicircular families, {U } ∪ {UN }N ∈N , respectively. Recall that the integer-shift group B of Eq. (3.6.11) acts naturally on the semicircular filterization LQ , and hence, the corresponding group-dynamical system (3.7.6), ΓQ = (B, LQ , α) , is well-determined, and it induces the corresponding crossed product Banach ∗-probability space, LQ = (LQ , τo ) ,

the dynamical semicircular ∗-probability space (3.7.10). Also see Refs. [1,3–5,7–12] for more motivations. In this section, different from the dynamics of Section 3.7, we concentrate on the actions of B as Banach-space operators on LQ by regarding LQ as a Banach space (e.g., [13,14]), and consider “restricted” actions of the integer-shift group B on the semicircular N -filterizations LQ [N ] of Eq. (3.8.2) for a fixed N ∈ N. DOI: 10.1201/9781003374817-4

141

142 ■ Constructive Analysis of Semicircular Elements

4.1

THE INTEGER-SHIFT-OPERATOR ALGEBRA B ON LQ

Let B = β k k∈Z be the integer-shift group acting on the semicircular filterization LQ , where the integer shifts β k are the free-isomorphisms on LQ . Now, let B (LQ ) be the operator space of Ref. [13] consisting of all bounded linear transformations on LQ , by regarding LQ as a Banach space. Since B ⊂ Aut (LQ ) , and since Aut (LQ ) ⊂ B (LQ ) , one can regard all integer shifts of B as Banach-space operators acting on LQ , i.e., B ⊂ B (LQ ) . (4.1.1) By Eq. (4.1.1), one can define the closed subspace B of B (LQ ) by B = C [B] in B (LQ ) ,

(4.1.2)

where Y are the operator-norm-topology closures of subsets Y of B (LQ ). Note that, since B is a group, all elements of the polynomial algebra C [B] are again contained in B (LQ ), the subspace B of Eq. (4.1.2) is well-defined in B (LQ ). Moreover, by definition, it forms a Banach algebra embedded in B (LQ ). Now, define a unary operation (∗) on the Banach algebra B of Eq. (4.1.2) by ∗

tk β k k∈Z

tk β −k , in B ,

=

(4.1.3)

k∈Z

where is the infinite (or, the limit of finite) sum(s), and z are the conjugates of z ∈ C, and where β −k are the group-inverses of β k . Then it is not difficult to check that the operation (∗) of Eq. (4.1.3) is a well-defined adjoint on the Banach algebra B of Eq. (4.1.2). i.e., the embedded structure B is not only a closed subspace but also a well-defined Banach ∗-algebra in the operator space B (LQ ). Definition 4.1: Let B be the Banach ∗-algebra (4.1.2) equipped with its adjoint (4.1.3) in B (LQ ). Then we call B , the (integer-)shiftoperator algebra (on LQ ). All Banach-space operators of B are said to be (integer-)shift operators on LQ .

Certain Banach-Space Operators on LQ [N] ■ 143

As we have seen in Eq. (4.1.3), all shift operators T of the shiftoperator algebra B is expressed by tk β k , with tk ∈ C,

T =

k∈Z

having its adjoint,

(4.1.4)

T∗ =

tk β −k , with β k

∗

= β −1 .

k∈Z

By the definition (4.1.2), this shift-operator algebra B canonically acts on LQ by tk β k

tk β k (T ) ,

(T ) =

k∈Z

k∈Z

for all T ∈ LQ . Lemma 4.1: Let T ∈ B be a shift operator (4.1.4) on LQ , and let Uj ∈ U be a generating semicircular element of LQ . Then, for any n ∈ N, there exists ϖT = tk ∈ C, k∈Z

such that

(4.1.5) τ T Ujn

= ϖT ωn c n2 .

Proof . Let T ∈ B be a shift operator (4.1.4). Then, for any Uj ∈ U ⊂ LQ , one has T Ujn =

tk β k Ujn = k∈Z

n tk Uj+k , k∈Z

in LQ , for all n ∈ N. Thus, τ T Ujn

n tk τ Uj+k

= k∈Z

=

tk ωn c n2 , k∈Z

by the semicircularity of Uj +k ∈ U in LQ , for all k ∈ Z. Therefore, the free-distributional data (4.1.5) holds.

144 ■ Constructive Analysis of Semicircular Elements

It is interesting in the formula (4.1.5) that the quantity ϖT ∈ C is the sum of all coefficients of the shift operator T ∈ B . Now, let T ∈ B be a shift operator (4.1.4), and s ∈ N. Then s s

T =

tk β

k

s

s

=

tks β (k1 ,...,ks )∈Zs

k∈Z

= k ∈Z

kl l=1

l=1





  

 k tkl  β ,

s

(k1 ,...,ks )∈Zs ,

kl =k

(4.1.6)

l=1

in B , by the commutativity (3.6.12) on the generating group B of the shift-operator algebra B of Eq. (4.1.2). By Eq. (4.1.6), the more generalized result of Eq. (4.1.5) is obtained as follows. Theorem 4.1: Let T ∈ B be a shift operator (4.1.4), and s ∈ N, and let Uj ∈ U be a generating semicircular element of LQ . Then there exists   ϖT,s = k∈Z

   

s

(k1 ,...,ks )∈Zs ,

kl =k

  tkl  ∈ C, 

l=1

such that

(4.1.7) τ T s Ujn

= ϖT,s ωn c n2 ,

for all n ∈ N. Proof . By Eq. (4.1.6), the quantities ϖT,s ∈ C of Eq. (4.1.7) are the sums of all coefficients of the shift operators T s ∈ B , for all s ∈ N. So, for any arbitrary s, n ∈ N, the free-distributional data (4.1.7) holds by Eq. (4.1.5). The above theorem illustrates how our shift operators of B affect the original free probability on LQ . Different from the action of the integer-shift group B (which preserves the free probability on LQ ), the action of the shift-operator algebra B distorts the free probability on

Certain Banach-Space Operators on LQ [N] ■ 145

LQ in general by Eq. (4.1.7). Note that the quantity ϖT of Eq. (4.1.5) is understood to be ϖT,1 of Eq. (4.1.7) in C. In the rest of this section, we consider how the shift-operator B deforms the original free-distributional data on the semicircular filterization LQ by considering certain special cases of Eq. (4.1.7). Theorem 4.2: Let t ∈ C, and β k ∈ B, and let tβ k ∈ B be a shift operator. If Uj ∈ U is a generating semicircular element of LQ , then tβ k (Uj )

τ

n

= t n ωn c n2 ,

and

(4.1.8) tβ k

τ

∗

n

(Uj )

= tn ωn c n2 ,

for all n ∈ N. Proof . Under hypothesis, we have tβ k (Uj ) = tUj+k , and tβ k since β k

∗

∗

(Uj ) = tUj−k ,

= β −k in B , for all k ∈ Z, by Eq. (4.1.3). So, τ

tβ k (Uj )

n

n = tn τ Uj+k = tn ωn c n2 ,

and

(4.1.9) τ

tβ k

∗

n

(Uj )

n = tn τ Uj−k = tn ωn c n2 ,

for all n ∈ N. Therefore, the free-distributional data (4.1.8) holds by Eq. (4.1.9). The following corollary is a direct consequence of the above theorem. Corollary 4.1: Let t ∈ R× in C, and β k ∈ B, and let T = tβ k ∈ B be the corresponding shift operator. If Uj ∈ U is a generating semicircular element of LQ , then T (Uj ) is t 2 -semicircular in LQ . Proof . Let T = tβ k ∈ B be the shift operator for t ∈ R× . Then the image T (Uj ) = tUj+k ∈ LQ

146 ■ Constructive Analysis of Semicircular Elements

is self-adjoint in LQ , and hence, the free distribution of it is characterized by the free moments, n

τ ((T (Uj )) ) = ωn tn c n2 = ωn t2

n 2

c n2 ,

for all n ∈ N, by Eq. (4.1.8). So, by Eq. (2.7.5), this self-adjoint free random variable T (Uj ) is t 2 -semicircular in LQ . The above corollary shows that the shift operator tβ k ∈ B with t ∈ R× and β k ∈ B deform the semicircular law induced by U to the t 2 -semicircular law on LQ . Theorem 4.3: Let j1 = ̸ j2 in Z, and Uj1 , Uj2 ∈ U , the corresponding generating semicircular elements of LQ , and let √ 1 W = √ (Uj1 + iUj2 ) ∈ LQ , with i = −1, 2 be a circular element of LQ . If T = tβ k ∈ B is a shift operator with t ∈ C× and β k ∈ B, then the free distribution of T (W ) is characterized by the only nonzero joint free cumulants, 2

k2 (T (W )∗ , T (W )) = |t| = k2 (T (W ), T (W )∗ ) , or, by the only nonzero joint free moments, n

τ ((T (W )∗ T (W )) ) = |t|

2n

(4.1.10) n

|X (2n)| = τ ((T (W )T (W )∗ ) ) ,

of {T (W ), T (W )∗ }, for all n ∈ N, where |t| is the modulus of t. Proof . By Theorem 3.7, the operator W = √12 (Uj1 + iUj2 ) is the circular element of LQ generated by the two, free semicircular elements Uj1 , Uj2 ∈ U . Observe that t T (W ) = √ (Uj1 +k + iUj2 +k ) = tTk , 2 where

1 Tk = √ (Uj1 +k + iUj2 +k ) ∈ LQ 2 is the circular element of LQ generated by the two, free semicircular elements Uj1 +k , Uj2 +k ∈ U because j1 + k ̸= j2 + k in Z

Certain Banach-Space Operators on LQ [N] ■ 147

(for all k ∈ Z), by Theorem 3.7. Since t ∈ C× , the free distribution of T (W ) is characterized by the only nonzero joint free cumulants of tTk , tTk∗ , 2 2 k2 tTk , tTk∗ = |t| k2 (Tk , Tk∗ ) = |t| , and 2

k2 tTk∗ , tTk = |t| , by Eq. (1.3.8), or, by the only nonzero joint free moments of them, τ |t|

2n

(Tk Tk∗ )

n

= |t|

τ |t|

2n

(Tk∗ Tk )

2n

|X2 (2n)| ,

n

= |t|

2n

|X2 (2n)| ,

and for all n ∈ N, by Eq. (1.3.13). Thus, the free-distributional data (4.1.10) holds. The above theorem illustrates how a shift operator tβ k ∈ B with t ∈ C× and β k ∈ B deform the circular law on LQ by Eq. (4.1.10). Theorem 4.4: Let j1 = ̸ j2 in Z, and Uj1 , Uj2 ∈ U , the corresponding generating semicircular elements of LQ , and let W = Uj1 Uj2 Uj1 ∈ LQ be a free Poisson element. If T = tβ k ∈ B is a shift operator with t ∈ R× and β k ∈ B, then the free random variable T (W ) ∈ LQ is a free Poisson element whose free distribution is characterized by the free cumulants, n kn (T (W ), ..., T (W )) = ωn t2 2 c n2 , or, by the free moments,

(4.1.11) 



n

τ ((T (W )) ) = ωn tn  π∈N Ce (n)

c |V |  , V ∈π

2

for all n ∈ N. Proof . Indeed, by Corollary 3.2, the given free random variable W is a free Poisson element of LQ . Under hypothesis, the free random variable, T (W ) = tUj1 +k Uj2 +k Uj1 +k = Uj1 +k (tUj2 +k ) Uj1 +k

148 ■ Constructive Analysis of Semicircular Elements

is a free Poisson element of LQ too by Theorem 3.8 because j1 + k ̸= j2 + k in Z, implying the freeness of Uj1 +k and tUj2 +k in LQ . Remark that, since t ∈ R× , the operator tUj2 +k is self-adjoint in LQ . Since T (W ) is a free Poisson element of LQ , the free distribution of T (W ) is characterized by the free cumulants, n

kn (T (W ), ..., T (W )) = τ ((tUj2 +k ) ) = tn ωn c n2 , or, by the free moments, n

ω|V | t|V | c |V |

τ ((T (W )) ) = π∈N C(n)

V ∈π

,

2

for all n ∈ N, by Theorem 3.8. So, the formulas (4.1.11) hold. The above theorem characterizes how our shift operators tβ k ∈ B with t ∈ R× and β k ∈ B deform the free Poisson distributions induced by U on LQ by Eq. (4.1.11). We finish this section with the following general result. tk β k ∈ B be an arbitrary shift operator

Theorem 4.5: Let T = k ∈Z

(4.1.4), and let the generating semicircular elements Uj1 , ..., UjN ∈ U , for N ∈ N, generate the free (reduced or non-reduced) word, N

Ujl ∈ LQ ,

U = l=1

where j1 , ..., jN are not necessarily distinct in Z. If τ (U ) = wU ∈ C is characterized by Eq. (3.5.1), then there exist the quantities ϖT,s ∈ C of Eq. (4.1.7), such that τ (T s (U )) = ϖT,s wU ,

(4.1.12)

for all s ∈ N. Proof . Assume that U ∈ LQ is the above free random variable satisfying τ (U ) = wU in C, by Eq. (3.5.1) (or, by Eq. (1.4.13)). Since

Certain Banach-Space Operators on LQ [N] ■ 149

all integer shifts β k ∈ B are free-isomorphisms on LQ by Eq. (3.6.10), we have τ β k (U ) = wU = τ (U ) , for all k ∈ Z. Recall that, by Eq. (4.1.6),  Ts = k∈Z

   



s

(k1 ,...,ks )∈Zs ,

kl =k

  tkl  β k , 

l=1

in B , and hence,  T s (U ) = k∈Z



   

s

(k1 ,...,ks )∈Zs ,

kl =k

  tkl  β k (U ) , 

l=1

in LQ . Thus, τ (T s (U )) = ϖT,s wU , ∀s ∈ N. Therefore, the free-distributional data (4.1.12) holds. In this section, we studied how our shift-operator algebra B deform the free probability on the semicircular filterization LQ . In the following section, let’s consider how the restrictions of shift operators of B act on the semicircular N -filterizations LQ [N ] of Eq. (3.8.3), for N ∈ N.

4.2

RESTRICTED-SHIFT-OPERATOR ALGEBRAS BN ON LQ [N ]

Fix N ∈ N throughout this section, and let LQ [N ] be the semicircular N -filterization (3.8.3), generated by the free semicircular family UN = N {Uj }j=1 (as a free-probabilistic sub-structure of the semicircular filterization LQ , representing all Banach ∗-probability spaces generated by mutually free N -many semicircular elements, whose unities are identical free-distributed to be τ (1N )). In this section, we consider the restricted group-action of our integer-shift group B (or the restricted algebra-action of the shiftoperator algebra B ) on LQ [N ], and study how such restricted actions deform the original free-distributional data on LQ [N ].

150 ■ Constructive Analysis of Semicircular Elements

Let B be the integer-shift group acting on the semicircular filterization LQ . Now, act B on the semicircular N -filterization LQ [N ] by an action αN , αN β k

denote

=

k βN ∈ Hom (LQ [N ]) ,

defined by the morphism satisfying  k  β (Uj ) = Uj+k k βN (Uj ) =  0N

(4.2.1) if Uj+k ∈ UN otherwise,

for all k ∈ Z, where 0N = 0LQ |LQ [N ] is the zero operator of LQ [N ], N where UN = {Uj }j=1 is the free N -semicircular family generating LQ [N ] (understood to be a sub-family of the free semicircular family U generating LQ ), where Hom (LQ [N ]) = {η : η is a ∗ -homomorphism on LQ } , is the homomorphism semigroup of LQ [N ], under the multiplication (or the composition) on ∗-homomorphisms. By the definition (4.2.1) of the restricted action αN of B, one has ∗-homomorphisms,  if 1 ≤ j + k ≤ N  Uj+k k k αN β (Uj ) = βN (Uj ) = (4.2.2)  0N otherwise, for all k ∈ Z, for all j ∈ {1 , ..., N }. Note that, since β k ∈ B are ∗-isomorphisms on LQ , the restrictions k αN β k = βN are well-defined ∗-homomorphisms on LQ [N ], satisfying (4.2.2). Note also that, since B is an abelian subgroup of the automorphism group Aut (LQ ), for any β k1 , β k2 ∈ B, we have β k1 β k2 = β k1 +k2 = β k2 +k1 = β k2 β k1 , in B.

(4.2.3)

Observe however that, under the restricted action αN of Eq. (4.2.1), if j ∈ {1 , ..., N } , with 1 ≤ 2j ≤ N ,

Certain Banach-Space Operators on LQ [N] ■ 151

then −j j βN βN (Uj ) = β −j (U2j ) = U2j−j = Uj ,

while,

(4.2.4) β j β −j (Uj ) = β j (Uj−j ) = β j (0N ) = 0N ,

in LQ , by Eq. (4.2.2). Proposition 4.1: Let αN be the restricted action (4.2.1) of B on LQ [N ]. Then, for k1 , k2 ∈ Z, k1 k2 k2 k1 βN βN ̸= βN βN , in Hom (LQ [N ]) ,

and

(4.2.5) k1 k2 βN βN ∈ / αN (B) ,

in general. The equality, and the inclusion of Eq. (4.2.5) hold, if and only if either k1 = 0 , or k2 = 0 in Z. Proof . First, assume that either k1 = 0 , or k2 = 0 in Z. Say k1 = 0 in Z. Then, for any k2 ∈ Z, k1 k2 k2 k2 0 k2 k1 0 k2 βN βN = βN βN = βN = βN βN = βN βN ,

in Hom (LQ [N ]), since

(4.2.6)

0 βN = αN β 0 = IN , on LQ [N ],

by Eq. (4.2.2), where IN = 1LQ [N ] = 1LQ |LQ [N ] is the identity map on LQ [N ] (contained in Hom (LQ [N ])). So, by Eq. (4.2.6), if either k1 = 0 , or k2 = 0 in Z, then the equality, and the inclusion of Eq. (4.2.5) hold. Conversely, if neither k1 = 0 , nor k2 = 0 in Z, then the equality, and the inclusion of Eq. (4.2.5) do not hold by Eq. (4.2.4). Therefore, the equality, and the inclusion of Eq. (4.2.5) hold, if and only if either k1 = 0 , or k2 = 0 in Z. Therefore, the relations of Eq. (4.2.5) holds in general. The above proposition shows that, different from (4.2.3), the subset, αN (B) ⊂ Hom (LQ [N ]) ,

152 ■ Constructive Analysis of Semicircular Elements

does not have a nice algebraic structure inherited from that of the homomorphism semigroup Hom (LQ ). i.e., even though integershift group B is a subgroup of Aut (LQ ), the subset αN (B) of the restrictions does not form an algebraic sub-structure of Hom (LQ [N ]) by Eq. (4.2.5), whenever N ∈ N. It means that, different from Sections 3.7 and 4.1, we have neither a suitable algebraic-structure-depending dynamical system of αN (B) acting on LQ [N ], nor a suitable algebraic-structure-depending algebra in the operator space B (LQ [N ]), consisting of all bounded linear transformations on the Banach space LQ [N ]. However, one can define a Banach algebra generated by the set αN (B), since αN (B) ⊂ Hon (LQ [N ]) ⊂ B (LQ [N ]) . Definition 4.2: The subset, k αN (B) = αN β k = βN : βk ∈ B ,

(4.2.7)

of the operator space B (LQ [N ]) is called the restricted-shift family k (acting) on LQ [N ], or in B (LQ [N ]). All elements βN are said to k∈N be restricted(-integer) shifts on LQ [N ]. For convenience, we denote the restricted-shift family αN (B) by BN . Even though the relations of Eq. (4.2.5) hold, all products of restricted shifts of BN are ∗-homomorphisms on LQ [N ], and hence, they are contained in B (LQ [N ]). So, from the restricted-shift family BN of Eq. (4.2.7), one can define a topological subset, def

BN = C [BN ] in B (LQ [N ]) ,

(4.2.8)

where Z are the operator-norm-topology closures of subsets Z of B (LQ [N ]). By Eqs. (4.2.5) and (4.2.8), this subset BN of Eq. (4.2.8) is a welldefined Banach algebra embedded in B (LQ [N ]). By definition, every element T ∈ BN is expressed by   ∞

T =

n

 n=0

(k1 ,...,kn )∈Zn

in BN , where t(k1 ,...,kn ) ∈ C.

kl  βN ,

t(k1 ,...,kn ) l=1

Certain Banach-Space Operators on LQ [N] ■ 153

Define now a unary operation (∗) on BN by ∞

n

t(k1 ,...,kn ) n=0

(k1 ,...,kn )∈Zn def

l=1

∗ kl βN

∞

n

=

t(k1 ,...,kn ) n=0

(k1 ,...,kn )∈Zn

l=1

−kn−l+1

βN

.

(4.2.9)

By Eq. (4.2.9), one can get that k1 k2 kn βN βN ...βN

∗

−kn −k2 −k1 = βN ...βN βN ,

for all n ∈ N. So, it is not difficult to check that this unary operation (4.2.9) is a well-defined adjoint on the Banach algebra BN of Eq. (4.2.8), since ∗

T ∗∗ = T, and (tT ) = tT ∗ ; and ∗

(T1 + T2 ) = T1∗ + T2∗ ; and ∗

(T1 T2 ) = T2∗ T1∗ , in BN , for all t ∈ C, and T , T1 , T2 ∈ BN . Proposition 4.2: Let BN be the Banach algebra (4.2.8) embedded in B (LQ [N ]). Then it forms a Banach ∗-algebra in B (LQ [N ]) with its adjoint (4.2.9). Equivalently, Every operator T of BN is adjointable (in the sense of Ref. [13]) in B (LQ [N ]) with its unique adjoint T ∗ . Proof . Since the unary operation (4.2.9) is a well-defined adjoint, the Banach algebra BN of Eq. (4.2.8) forms a well-defined Banach ∗-algebra in B (LQ [N ]). The above proposition shows that, even though the restrictedshift family BN does not have a nice algebraic sub-structure in Hom (LQ [N ]), it generates a well-defined Banach ∗-algebra BN in the operator space B (LQ [N ]), acting on the semicircular N -filterization LQ [N ]. Definition 4.3: The Banach ∗-algebra BN of Eq. (4.2.8) is called the (restricted-)shift-operator algebra on LQ [N ] (or, in B (LQ [N ])). All operators of BN are said to be (restricted-)shift operators on LQ [N ].

154 ■ Constructive Analysis of Semicircular Elements

In the rest of this section, we study the generator set BN , the restricted-shift family, of the shift-operator algebra BN . Theorem 4.6: Set-theoretically, the restricted-shift family BN is identical to k BN = βN

N k=−N

−N N , with βN = ON = βN ,

(4.2.10)

where ON ∈ Hom (LQ [N ]) is the zero map, and hence, the zero operator on LQ [N ]. k Proof . Show that βN = ON in Hom (LQ [N ]), if and only if k ≥ N , or k ≤ −N in Z. Assume first that

k ≥ N, or k ≤ −N, in Z. Then β k (Uj ) = Uj+k in LQ , with j + k > N, respectively, j + k < −N, in Z, implying that k βN (Uj ) = 0N , in LQ [N ],

for all j ∈ {1 , ..., N } by Eq. (4.2.2), showing that k βN = ON in Hom (LQ [N ]) .

Conversely, suppose that 1 < k < N, in Z. Then, for such k , there always exists at least one j ∈ {1 , ..., N }, such that 1 ≤ j + k ≤ N, equivalently, k βN (Uj ) = Uj+k ∈ UN ,

in LQ [N ] by Eq. (4.2.2), and hence, k βN ̸= ON , in Hom (LQ [N ]) .

Certain Banach-Space Operators on LQ [N] ■ 155 k Therefore, βN = ON in Hom (LQ [N ]), if and only if

k ≥ N, or k ≤ −N, in Z. i.e., set-theoretically, the restricted-shift family BN satisfies that k BN = αN (B) = βN : k = 0, ±1, ..., ±N , −N 0 N with the identity map βN = IN , and the zero map βN = ON = βN . i.e., the set-equality (4.2.10) holds true.

The above theorem shows that every nonzero shift operator T ∈ BN actually has its expression,   ∞

T =

n

n=0

kl  βN ,

t(k1 ,...,kn )

 (k1 ,...,kn )∈{0,±1,...,±(N −1)}n

l=1

in B (LQ [N ]), by Eq. (4.2.10).

4.3

FREE PROBABILITY ON LQ [N ] DEFORMED BY BN

In this section, we study how our (restricted-)shift-operator algebra BN of Eq. (4.2.8), generated by the restricted-shift family, k BN = βN

N k=−N

−N N , with βN = ON = βN ,

deform the original free-distributional data on the semicircular N -filterization LQ [N ], for a fixed N ∈ N. We show that, even though the integer shifts β k ∈ B are freek isomorphisms on LQ , the restricted shifts βN = αN β k do not preserve the free probability on LQ [N ], in general, by Eq. (4.2.1) and k (4.2.2). More precisely, the restricted shifts βN ∈ BN , with k ̸= 0 in {0, ±1, ..., ±N }, do not preserve the free probability on LQ [N ]. k Lemma 4.2: Let βN ∈ BN be a generating shift operator of BN , for k ∈ {0 , ±1 , ..., ±N }. 0 If k = 0 , then βN is a free-isomorphism on LQ [N ].

(4.3.1)

k If k ̸= 0 , then βN is not a free-homomorphism. In particular, it satisfies (4.3.2)

156 ■ Constructive Analysis of Semicircular Elements

τN

k βN

(Uj )

n

=

  ωn c n2 = τ Ujn 

if 1 ≤ j + k ≤ N

0 ̸= τ Ujn

otherwise,

for all n ∈ N, for Uj ∈ UN , for all j ∈ {1 , ..., N }. 0 Proof . Suppose k = 0 , and hence, βN = αN β 0 ∈ BN is identified with the identity map IN on LQ [N ]. Then it is not only a ∗-isomorphism but also a free-isomorphism on LQ [N ]. i.e., the statement (4.3.1) holds. k ̸ 0 in {0, ±1, ..., ±N }, and βN ̸= IN in BN ⊂ Assume now that k = BN . Then, for any generating semicircular element Uj ∈ UN of LQ [N ], one has that  n  τN Uj+k if 1 ≤ j + k ≤ N  n k τN βN (Uj ) =   τN (0N ) otherwise,

for all n ∈ N, implying the free-distributional data in the statement (4.3.2). i.e., the statement (4.3.2) holds. The above lemma shows how the action of the shift-operator algebra BN deforms the semicircular law on LQ [N ] induced by UN , by Eqs. (4.3.1) and (4.3.2). Theorem 4.7: Let Uj1 , ..., UjN ∈ UN be the generating semicircular elements of LQ [N ], for j1 , ..., jN ∈ {1, ..., N } , which are not necessarily distinct from each other, and let N

Ujl ∈ LQ [N ] be the corresponding free (reduced or non-

U = l=1

reduced) word. If τN (U ) = wU , in C, k characterized by Eq. (1.4.13), and if βN ∈ BN is a generating shift operator of BN , for

k ∈ {0, ±1, ..., ±N } ,

Certain Banach-Space Operators on LQ [N] ■ 157

then

(4.3.3)

τN β k (U ) =

    wU   

if 1 ≤ jl + k ≤ N for ”all” l = 1, ..., N

0

otherwise.

Ujl +k

if 1 ≤ jl + k ≤ N for all l = 1, ..., N

Proof . Under hypothesis,

k βN (U ) =

   

N l=1

  

0N

otherwise,

in LQ [N ], by Eqs. (4.2.1) and (4.2.2). Therefore, by Eqs. (4.3.1) and (4.3.2), the free-distributional data (4.3.3) is obtained. The above theorem characterizes how the generating shift operators of BN ⊂ BN deform the general free-distributional data on the semicircular N -filterization LQ [N ]. Define now a self-adjoint shift operator R ∈ BN by an operator, N −1

N

k βN ∈ BN ,

k βN =

R=

(4.3.4)

k=−(N −1)

k=−N

N

induced by all generating shift operators BN = β k k=−N . (Remark ±N that since βN = ON , the second equality of Eq. (4.3.4) holds.) Indeed, this shift operator R of Eq. (4.3.4) is self-adjoint in BN , since ∗

N ∗

k βN

R =

N

k=−N

N −k βN =

= k=−N

l βN = R,

l=−N

in BN , by Eq. (4.2.9). Observe now that, if Uj ∈ UN is a generating semicircular element of LQ [N ], for j ∈ {1 , ..., N }, then N

k =−N

N −j

N k βN

R (Uj ) =

(Uj ) = k =−N

k βN (Uj ) = l=1−j

Uj+k ,

(4.3.5)

158 ■ Constructive Analysis of Semicircular Elements

in LQ [N ] by Eq. (4.2.2) because the summands satisfy k βN (Uj ) = Uj+k ̸= 0N ⇐⇒ 1 ≤ j + k ≤ N,

if and only if

(4.3.6) 1 − j ≤ k ≤ N − j,

for k ∈ {0 , ±1 , ..., ±N }. N

k βN ∈ BN be the shift operator (4.3.4).

Lemma 4.3: Let R = k =−N

If Uj ∈ UN is a generating semicircular element of LQ [N ], for j ∈ {1 , ..., N }, then N −j

R (Uj ) =

Uj +k , in LQ [N ].

(4.3.7)

k =1 −j

Proof . By the relation (4.3.6), the operator-equalities (4.3.5) are satisfied, which implies the formula (4.3.7). By the above lemma, one obtains the following deformed semicircularity on the semicircular N -filterization LQ [N ] by the action of the shift operator R ∈ BN of Eq. (4.3.4). Theorem 4.8: Let R ∈ BN be the shift operator (4.3.4), and let Uj ∈ UN be a generating semicircular element of LQ [N ], for j ∈ {1 , ..., N }. Then τ R Ujn

= N ωn c n2 ,

(4.3.8)

for all n ∈ N. N

k βN ∈ BN be the shift operator (4.2.4)

Proof . Let R = k=−N

induced by the restricted-shift family BN , and let Uj ∈ UN be a given semicircular element of LQ [N ]. Then N −j

N

R Ujn = k=−N

k βN Ujn = k=1−j

n Uj+k ,

Certain Banach-Space Operators on LQ [N] ■ 159

in LQ [N ] by Eq. (4.3.7), since BN ⊂ Hom (LQ [N ]) in BN , for all n ∈ N. So,   N −j

τ R Ujn

=

N −j

ωn c n2 = ωn c n2  k=1−j

1 ,

k=1−j

by the semicircularity of Uj +k ∈ UN in LQ [N ], for all k = 1 − j , ..., 0 , ..., N − j , for j ∈ {1 , ..., N }. It is easy to check that N −j

1 = |{1 − j, ..., 0, ..., N − j}| k=1−j

= |{1 − j, ..., −1}| + |{0}| + |{1, ..., N − j}| where |Z| are the cardinalities of sets Z = (j − 1 ) + 1 + (N − j ) = N, for all j ∈ {1 , ..., N }, implying that τ R Ujn

= N ωn c n2 , ∀n ∈ N.

Therefore, the free-distributional data (4.3.8) holds. The above theorem illustrates how the shift operator R ∈ BN deforms the semicircular law induced by the generating free semicircular family UN of the semicircular N -filterization LQ [N ] by Eq. (4.3.8).

4.4

CONDITIONAL-NILPOTENT-PROPERTY ON BN

Throughout this section, we fix N ∈ N, and the corresponding semicircular N -filterization LQ [N ] generated by the free semicircular N family UN = {Uj }j=1 . We here study a certain operator-theoretic N

property of the shift operators β k k=−N of the restricted-shift family BN , generating the shift-operator algebra BN . Especially we are interested in the conditional-nilpotent-property of them.

160 ■ Constructive Analysis of Semicircular Elements

Definition 4.4: Let Y be a Banach space with its zero vector 0Y , and its identity vector 1Y satisfying 1Y · y = y = y · 1Y , ∀y ∈ Y, and let B (Y ) be the operator space of all Banach-space operators on Y . A non-trivial operator T ∈ B (Y )\ {O, I} is said to be (C-) conditional-nilpotent, if there exists n ∈ N, such that T m = Oc in B (Y ) , ∀m ≥ n in N, where O is the zero operator, and I is the identity operator on Y , and  if y = t · 1Y , for t ∈ C  t · 1Y def Oc (y) =  0Y otherwise, for all y ∈ Y . In particular, if n0 ∈ N is the minimal quantity satisfying the above relation, then it is called the conditional-nilpotence of T . i.e., T m = Oc , ∀m ≥ n0 in N. By definition, one can verify that certain restricted shifts of BN are conditional-nilpotent as shift operators of BN . Theorem 4.9: If k ̸= 0 in {0 , ±1 , ..., ±N }, then the corresponding k generating shift operator βN ∈ BN is conditional-nilpotent in the shiftoperator algebra BN . Proof . Let Uj ∈ UN be an arbitrary generating semicircular element k of LQ [N ], and let βN ∈ BN be a restricted shift, generating BN , where ̸ 0 . By Eq. (4.2.1) and (4.2.2), we have k=  if 1 ≤ j + k ≤ N  Uj+k k βN (Uj ) =  0N otherwise, in LQ [N ], and hence, inductively,   Uj+nk k n βN (Uj ) =  0N for each n ∈ N.

if 1 ≤ j + nk ≤ N otherwise,

Certain Banach-Space Operators on LQ [N] ■ 161

So, by the Zorn’s lemma, there exists nk ,j ∈ N, such that j + nk,j k ≥ N in N, and hence,

(4.4.1) nk,j

k βN

(Uj ) = 0N , in LQ [N ].

Now take, nk = max {nk,j : j = 1, ..., N } ∈ N, where nk ,j ∈ N are in the sense of Eq. (4.4.1). Then one can take k βN

nk

implying that k βN

(Uj ) = 0N , ∀j ∈ {1, ..., N } ,

nk

(T ) = 0N , ∀T ∈ LQ [N ],

showing that

(4.4.2) k βN

nk

c = ON , in BN .

It shows that there exists at least one natural number nk ∈ N, such that k n c βN = ON , for all n ≥ nk , on LQ [N ] by Eq. (4.4.2), for all k ∈ {±1 , ..., ±N }. i.e., the generating k shift operators βN of BN are conditional-nilpotent in BN , whenever k= ̸ 0. Now, let k1 , ..., ks ∈ {0 , ±1 , ..., ±N } for s ∈ N (which are not necessarily distinct from each other), and let s def

β kl ∈ BN

W =

(4.4.3)

l=1

be a shift operator. Theorem 4.10: If kl = ̸ 0 , for some l = 1 , ..., s, then a shift operator W ∈ BN of Eq. (4.4.3) is conditional-nilpotent on LQ [N ]. Proof . If s = 1 , then it is proven by Theorem 4.9. So, we concentrate on the cases where s > 1 in N.

162 ■ Constructive Analysis of Semicircular Elements

If W ∈ BN is in the sense of (4.4.3), satisfying kl = ̸ 0, for some l ∈ {1, ..., s}, then there exists a quantity, nW = max {nkl : kl = ̸ 0 } ∈ N,

(4.4.4)

where nkl ∈ N are in the sense of Eq. (4.4.2), such that c W nW = ON , in BN .

Therefore, a shift operator W ∈ BN is conditional-nilpotent. The above theorem proves that all “finite” products of BN in BN \ 0 βN are conditional-nilpotent. Also, it shows the following result. Corollary 4.2: If an arbitrary shift operator  ∞

T = n=1

n

kl  βN ∈ BN

t(k1 ,...,kn )

 (k1 ,...,kn )∈{0 ,±1 ,...,±(N −1 )}n



l=1

0 has “finitely-many” nonzero summands with the “zero” βN -term (or the vanishing identity term), then it is conditional-nilpotent in BN .

Proof . Suppose a shift operator T ∈ BN has only finite nonzero 0 summands with the zero βN -term. i.e., s

tn Wn ∈ BN , for s ∈ N,

T = n=1

0 ̸ βN where tn ∈ C× , and Wn = ∈ BN are the finite products in the restricted shifts of BN . Then there exists a quantity, s

nT =

nWn ∈ N,

(4.4.5)

n=1 s

where the factors {nWn }n=1 are in the sense of Eq. (4.4.4), such that c T nT = ON , in BN ,

Certain Banach-Space Operators on LQ [N] ■ 163

and hence, the shift operator T ∈ BN is conditional-nilpotent on LQ [N ]. The above corollary shows that all “finite-sum” shift operators 0 T ∈ BN , with the vanishing βN -term, are conditional-nilpotent by the existence of the quantity nT ∈ N of Eq. (4.4.5). Theorem 4.11: Let T ∈ BN be a “finite-sum” shift operator, having 0 only finitely many nonzero summands with the zero βN -term, and let W ∈ LQ [N ] be an arbitrary free random variable. Then there exists n0 ∈ N, and there exists at least one ml ∈ N, such that ml ≥ n0 , then s

(T ml (W ))

τ

rl

= 0,

(4.4.6)

l=1 s

for (r1 , ..., rs ) ∈ {1, ∗} , and m1 , ..., ms ∈ N for all s ∈ N, whenever n ≥ n0 in N. 0 Proof . Let T ∈ BN be a finite-sum shift operator with the zero βN term. Then, by Corollary 4.2, there exists nT ∈ N of Eq. (4.4.5), such c that T nT = ON in BN . The existence of such a quantity nT ∈ N guarantees the existence of the minimal quantity n0 ∈ N, such that c T n0 = ON , in BN .

i.e., such a shift operator T is conditional-nilpotent in BN with its conditional-nilpotence n0 ∈ N. c It shows that, for all n ≥ n0 in N, T n = ON in BN . Thus, under hypothesis, the free-distributional data (4.4.6) holds. By Eq. (4.4.6), one obtains the following result. Corollary 4.3: Let T ∈ BN be a “finite-sum” shift operator with 0 the zero βN -term, and let W ∈ LQ [N ] be an arbitrary free random variable. Then there exists n0 ∈ N, such that the free random variables T n (W ) have their zero free distributions on LQ [N ], whenever n ≥ n0 in N. 0 Proof . Since T ∈ BN is a finite-sum shift operator with the zero βN term, it is conditional-nilpotent with its conditional-nilpotence n0 ∈ N.

164 ■ Constructive Analysis of Semicircular Elements

So, whenever n ≥ n0 in N, T n (W ) = 0N in LQ [N ], whose free distribution is the zero free distribution on LQ [N ]. The above theorem and corollary illustrate that the action of the shift-operator algebra BN (in particular, the actions of finite-sum shift 0 operators of BN with the zero βN -term) deforms original nonzero free distributions on LQ [N ] to the zero free distribution on it under the conditional-nilpotent-property on BN .

4.5

DEFORMED CIRCULAR LAWS ON LQ [N ] BY BN

In this section, by applying the main results of Sections 4.3 and 4.4, we consider how our action of the shift-operator algebra BN deform the circular law on the semicircular N -filterization LQ [N ] for a fixed N ∈ N. Throughout this section, let j1 ̸= j2 in {1, ..., N } , and

(4.5.1) 1 W = √ (Uj1 + iUj2 ) ∈ LQ [N ], 2

the corresponding circular element of LQ [N ], where Uj1 , Uj2 ∈ UN . k Lemma 4.4: Let βN ∈ BN be a restricted shift, a generating shift operator of BN , for k ∈ {0 , ±1 , ..., ±N }. If W ∈ LQ [N ] is a circular k element (4.5.1), then βN (W ) is circular in LQ [N ], if and only if

1 ≤ jl + k ≤ N, ∀l = 1, 2.

(4.5.2)

Proof . By Eq. (4.2.2), we have 1 k βN (W ) = √ (Uj1 +k + iUj2 +k ) , 2 k in LQ [N ], if and only if the condition (4.5.2) holds, if and only if βN (W ) is circular in LQ [N ] because

j1 + k ̸= j2 + k in {1, ..., N } ,

Certain Banach-Space Operators on LQ [N] ■ 165

implying the distinctness of Uj1 +k and Uj2 +k in the free semicircular family UN , guaranteeing the freeness of them in LQ [N ]. Also, one obtains the following lemma. k Lemma 4.5: Let βN ∈ BN be a generating shift operator of BN for k ∈ {0 , ±1 , ..., ±N }, and let W ∈ LQ [N ] be a circular element (4.5.1). k Then βN (W ) is 12 -semicircular in LQ [N ], if and only if

1 ≤ j1 + k ≤ N, and

(4.5.3) j2 + k < 1, or j2 + k > N.

Proof . Observe that the condition (4.5.3) is satisfied, if and only if 1 1 k βN (W ) = √ (Uj1 +k + i · 0N ) = √ Uj1 +k , 2 2 in LQ [N ] by Eq. (4.2.2), which is a self-adjoint free random variable, if and only if τ

k βN

(W )

n

=

1 √ 2

n

ωn c

n 2

= ωn

k for all n ∈ N, if and only if βN (W ) = 1 2

1 2

√1 Uj +k 1 2

n 2

c n2 , ∈ LQ [N ] is

-semicircular. Similarly, the following lemma is obtained.

k Lemma 4.6: Let βN ∈ BN be a generating shift operator of BN , for k ∈ {0 , ±1 , ..., ±N }, and let W ∈ LQ [N ] be a circular element (4.5.1). The condition; j1 + k < 1, or j1 + k > N,

and

(4.5.4) 1 ≤ j2 + k ≤ N ;

k holds, if and only if the free random variable −iβN (W ) ∈ LQ [N ] is 1 -semicircular. 2

166 ■ Constructive Analysis of Semicircular Elements

Proof . The condition (4.5.4) holds, if and only if −i 1 k −iβN (W ) = √ (0N + iUj2 +k ) = √ Uj2 +k , 2 2 in LQ [N ], as a self-adjoint free random variable, if and only if it is 1 2 -semicircular in LQ [N ], by Lemma 4.5. Suppose the condition (4.5.4) holds. Then W(k)

denote

=

i k βN (W ) = √ Uj2 +k ∈ LQ [N ], 2

(4.5.5)

which is not self-adjoint. So, the free distribution of this non-self-adjoint free random variable W(k ) ∈ LQ [N ] of Eq. (4.5.5) is characterized by ∗ the joint free moments, or by the joint free cumulants of W(k) , W(k) . n

Let (r1 , ..., rn ) ∈ {1, ∗} , for all n ∈ N. Consider that if W(k ) ∈ LQ [N ] is a free random variable (4.5.5), then n

τ l=1

rl W(k)

r1 r2 rn = τ W(k) W(k) ...W(k)

=

√i 2

=

√i 2

#(1 )

#(1)

−i √ 2 −i √ 2

#(∗)

τ Ujn2 +k

#(∗)

ωn c n2 ,

where

(4.5.6) #(1) = the number of 1’s in (r1 , ..., rn ) ,

and #(∗) = the number of ∗ ’s in (r1 , ..., rn ) . i.e., the free distribution of a free random variable W(k ) ∈ LQ [N ] of Eq. (4.5.5) is characterized by the joint free moments (4.5.6). k Lemma 4.7: Let βN ∈ BN be a generating shift operator of BN for k ∈ {0 , ±1 , ..., ±N }, and W ∈ LQ [N ], a circular element (4.5.1). The k condition (4.5.4) holds, if and only if the free distribution of βN (W ) ∈ LQ [N ] is characterized by the joint free moments,

Certain Banach-Space Operators on LQ [N] ■ 167 n

τ

k βN (W )

l=1 k k of βN (W ) , βN (W )

∗

rl

=

√i 2

#(1)

−i √ 2

#(∗)

ωn c n2 ,

(4.5.7)

n

, for (r1 , ..., rn ) ∈ {1 , ∗} , for all n ∈ N, where

#(1) = the number of 1’s in (r1 , ..., rn ) , and #(∗) = the number of ∗ ’s in (r1 , ..., rn ) . Proof . The condition (4.5.4) holds, if and only if i k βN (W ) = √ Uj2 +k ∈ LQ [N ], 2 by Eq. (4.5.5), if and only if it satisfies the joint free moments (4.5.7) by Eq. (4.5.6). Finally, one can get the following result. k Lemma 4.8: Let W ∈ LQ [N ] be a circular element (4.5.1), and βN ∈ BN , a generating shift operator of BN . Then the free distribution of k βN (W ) is the zero free distribution, if and only if

jl + k < 1, or jl + k > N, for all

(4.5.8) l = 1,2.

Proof . The condition (4.5.8) holds, if and only if k βN (W ) = 0N , in LQ [N ],

by Eqs. (4.2.1) and (4.2.2), if and only if the free distribution of it is the zero free distribution on LQ [N ]. By the above four lemmas, we have the following deformed circularity on LQ [N ] under the action of generating operators BN of BN . Theorem 4.12: Let W ∈ LQ [N ] be a circular element (4.5.1) and let k βN ∈ BN be a generating shift operator of BN .

168 ■ Constructive Analysis of Semicircular Elements k (W ) is the circular law, if and only (4.5.9) The free distribution of βN if 1 ≤ jl + k ≤ N, ∀l = 1, 2. k (4.5.10) The free distribution of βN (W ) is the and only if 1 ≤ j1 + k ≤ N,

1 2

-semicircular law, if

and j2 + k < 1, or j2 + k > N. k (4.5.11) The free distribution of βN (W ) is characterized by the joint free moments (4.5.7), if and only if

1 ≤ j2 + k ≤ N, and j1 + k < 1, or j1 + k > N. k (4.5.12) The free distribution of βN (W ) is the zero free distribution, if and only if jl + k < 1, or jl + k > N, ∀l = 1, 2.

Proof . The proof of Eqs. (4.5.9), (4.5.10), (4.5.11) and (4.5.12) are done by the Lemmas 4.4, 4.5, 4.7 and 4,8, respectively. The above theorem characterizes how the generating shift operators of BN , the restricted shifts of BN , deform the circular law (induced by UN ) on LQ [N ]. It shows that the circular law is deformed to be the circular law, or the weighted-semicircular law, or the free distribution characterized by the joint free moments (4.5.7), or the zero free distribution by the action of restricted-shift family BN .

4.6

DEFORMED FREE POISSON DISTRIBUTIONS ON LQ [N ] BY BN

In this section, we study how the free Poisson distributions on the semicircular N -filterization LQ [N ], induced by the generating free N semicircular family UN = {Uj }j=1 , are deformed by the action of the shift-operator algebra BN . In particular, we are interested in the cases

Certain Banach-Space Operators on LQ [N] ■ 169

where the shift operators are the generating ones, the restricted shifts k N of BN = βN . k=−N For j1 = ̸ j2 in {1, ..., N }, define a free Poisson element, W = Uj1 Uj2 Uj1 ∈ LQ [N ],

(4.6.1)

whose free distribution is characterized by knN (W, ..., W ) = τ Ujn2 = ωn c n2 , or, by

(4.6.2) 



τN (W n ) = ωn  π∈N Ce (n)

c |V | V ∈π

2

,

by Corollary 3.2, for all n ∈ N, where k•N (...) is the free cumulant on LQ [N ] in terms of the linear functional τN , and NCe (n) = {θ ∈ N C(n) : ∀V ∈ θ, |V | is even} in the noncrossing-partition lattice NC (n) for n ∈ N. Lemma 4.9: Let W ∈ LQ [N ] be a free Poisson element (4.6.1), and k k βN ∈ BN , a generating shift operator of BN . Then βN (W ) is a free Poisson element of LQ [N ], if and only if 1 ≤ jl + k ≤ N, ∀l = 1, 2.

(4.6.3)

k And the free Poisson distribution of βN (W ) is identical to that of W , characterized by Eq. (4.6.2).

Proof . The condition (4.6.3) holds, if and only if k βN (W ) = Uj1 +k Uj2 +k Uj1 +k ∈ LQ [N ],

by Eqs. (4.2.1) and (4.2.2), if and only if it is a free Poisson element of LQ [N ], since j1 + k ̸= j2 + k in {1, ..., N } .

170 ■ Constructive Analysis of Semicircular Elements

So, the corresponding free Poisson distribution is determined by Eq. (4.6.2), by the universality of the semicircular law (e.g., see Corollary 3.2). Lemma 4.10: Let W ∈ LQ [N ] be a free Poisson element (4.6.1), and k let βN ∈ BN be a generating shift operator of BN . Then the condition k (4.6.3) is not satisfied, if and only if βN (W ) is the zero operator of LQ [N ] whose free distribution is the zero free distribution. Proof . Suppose the condition (4.6.3) is not satisfied, if and only if either j1 + k < 1, or j1 + k > N, or j2 + k < 1, or j2 + k > N, if and only if k βN (W ) = 0N , in LQ [N ],

if and only if the free distribution of it is the zero free distribution on LQ [N ]. By the above two lemmas, we obtain the following theorem. Theorem 4.13: Let W ∈ LQ [N ] be a free Poisson element (4.6.1) k induced by the generating free semicircular family UN , and let βN ∈ BN be a generating shift operator of BN . Then the free random k variable βN (W ) is either a free Poisson element, whose free distribution is identical to that of W , or the zero element of LQ [N ], whose free distribution is the zero free distribution on LQ [N ]. Proof . It is proven by Lemmas 4.9 and 4.10.

4.7

THE SEMICIRCULAR ∞-FILTERIZATION LQ [∞]

In this section, we study, to some extent, different free-probabilistic sub-structure of the semicircular filterization LQ generated by the free semicircular family U = {Uj }j ∈Z . In Sections 4.2–4.6, we studied the free-probabilistic sub-structures, the semicircular N -filterizations LQ [N ] of LQ , generated by the “finite” free semicircular families UN = N {Uj }j=1 . Here, we are interested in a free-probabilistic sub-structure,

Certain Banach-Space Operators on LQ [N] ■ 171

LQ [∞] = (LQ [∞], τ∞ ) of LQ ,

(4.7.1)

generated by the free semicircular sub-family of U , ∞

U∞ = {Uj }j=1 , with ∞ = |N| .

Similar to Eq. (4.2.7), one can define a restricted action α∞ of the integer-shift group B on LQ [∞] by the ∗-homomorphism, satisfying α∞ β k (Uj )

denote

=

k β∞ (Uj ) =

  Uj+k 

0∞

if j + k ∈ N (4.7.2) otherwise,

for all j ∈ N. Then construct the restricted-shift family, def

B∞ = α∞ (B) = α∞ β k

denote

=

k β∞ : βk ∈ B ,

(4.7.3)

and the corresponding (restricted-)shift-operator algebra, def

B∞ = C [B∞ ], in the operator space B (LQ [∞]),

(4.7.4)

as in Eq. (4.2.8), where LQ [∞] is the semicircular ∞-filterization (4.7.1). As in Eq. (4.7.2), we have that k1 k2 k2 k1 β∞ β∞ = ̸ β∞ β∞ ,

and

(4.7.5) k1 k2 β∞ β∞ ∈ / B∞ ,

in general, in the homomorphism semigroup Hom (LQ [∞]), or, in the shift-operator algebra B∞ of Eq. (4.7.4), for k1 , k2 ∈ Z. However, different from the cases where N < ∞ in N, one obtains the following result in the case where N = ∞. k Lemma 4.11: If β∞ ∈ B∞ is a generating shift operator for k ∈ Z, then it is nonzero in B∞ . Equivalently, the following set-equality holds; k k c B∞ = β∞ : k ∈ Z , with β∞ = ̸ O∞ , ∀k ∈ Z,

(4.7.6)

172 ■ Constructive Analysis of Semicircular Elements c where O∞ is the operator of B∞ ⊂ B (LQ [∞]) in the sense of Definition 4.4 (by replacing N to ∞).

Proof . For any arbitrarily fixed k ∈ Z, there always exists at least one j ∈ N, such that j + k ≥ 1 ⇐⇒ j + k ∈ N, by the Zorn’s lemma (or, the axiom of choice for N ⊂ Z). It shows k that, for β∞ = α∞ β k ∈ B∞ , for all k ∈ Z, there always exists a generating semicircular element Uj ∈ U∞ of LQ [∞], such that k β∞ (Uj ) = Uj+k ̸= 0∞ ,

in LQ [∞], implying that k c β∞ in Hom (LQ [∞]) , ̸= O∞

and hence, k c β∞ in B∞ , ∀k ∈ Z. ̸= O∞

Therefore, set-theoretically, one has the equality (4.7.6); k B∞ = β∞

k∈Z

in Hom (LQ [∞]) ,

consisting of all nonzero elements. The above lemma provides a difference between B∞ and {BN }N ∈N . However, symbolically, all restricted-shift families can be expressed by k BN = βN

N k=−N

def

, ∀N ∈ N∞ = N ∪ {∞} .

All main results of Sections 4.2, 4.3, 4.5 and 4.6 are similarly applicable to the case of the shift-operator algebra B∞ of Eq. (4.7.4), acting on the semicircular ∞-filterization LQ [∞] of Eq. (4.7.1). However, by Eq. (4.7.6), the conditional-nilpotent-property of Section 4.4 does not hold in this case. k Theorem 4.14: Let β∞ ∈ B∞ be a generating shift operator of B∞ for k ∈ Z. Then it is not conditional-nilpotent in B∞ . 0 Proof . If k = 0 in Z, then β∞ = I∞ , the identity operator on LQ [∞] in B∞ (or, in B (LQ [∞])), and hence, it is not conditional-nilpotent in B∞ .

Certain Banach-Space Operators on LQ [N] ■ 173 k Assume now that k = ̸ 0 in Z, and β∞ ∈ B∞ . Then, by the proof of Eq. (4.7.6), there always exists at least one generating semicircular element Uj ∈ U∞ of LQ [∞], such that k β∞ (Uj ) = Uj+k ∈ U∞ ,

in LQ [∞], implying that n

k β∞

(Uj ) = Uj+nk ∈ U∞ ,

in LQ [∞], for all n ∈ N. It shows that k β∞

n

c in B∞ , ∀n ∈ N, ̸= O∞

for all k ∈ Z \ {0 }, too. Therefore, every generating shift operator k β∞ ∈ B∞ is not conditional-nilpotent in B∞ , for all k ∈ Z. As we mentioned above, even though there is no conditionalnilpotent-property on B∞ , the main results of Sections 4.2, 4.3, 4.5 and 4.6 are well-applicable to the B∞ -case. For instance, the deformed freedistributional data of the sections are similarly obtained if we replace N to ∞. For example, one obtains the following result. Theorem 4.15: If R = k ∈Z

k tk β∞ ∈ B∞ is a shift operator with tk ∈ C,

and if Uj ∈ U∞ is a generating semicircular element of LQ [∞], then ∞

τ∞ R Ujn

= ωn c n2

tl ,

(4.7.7)

l=1−j

for all n ∈ N. Proof . Suppose R ∈ B∞ is a given shift operator. Then ∞−j

R

Ujn

=

n tl Uj+l ∈ LQ [∞],

l=1−j

by Eq. (4.7.2), for all n ∈ N. ∞ Thus, by the semicircularity of {Uj+l }l=1−j ⊂ U∞ , the freedistributional data (4.7.7) holds. Readers can compare (4.7.7) with (4.3.8).

174 ■ Constructive Analysis of Semicircular Elements

Observation 4.1: All main results of Sections 4.2, 4.3, 4.5 and 4.6 are similarly obtained even if we replace N to ∞. However, by Theorem 4.14, the conditional-nilpotent-property of Section 4.4 does not hold.

4.8

MORE ABOUT LQ [∞]

This last subsection, Section 4.8, is somewhat independent from the main purposes of Section 4. So, readers may/can regard this section as an independent additional topic from the previous Sections 4.1– 4.7. However, the main results provide interesting free-homomorphic relations. In this section, we concentrate on studying free-probabilistic connections between the semicircular ∞-filterization LQ [∞] and the semicircular filterization LQ . In particular, we consider LQ [∞] in two aspects. The first one is a natural approach by understanding LQ [∞] as a free-homomorphic sub-structure of LQ by the very definition (4.7.1). The other one is to understand it as a free-isomorphic structure with LQ . Theorem 4.16: The semicircular ∞-filterization LQ [∞] is freehomomorphic to the semicircular filterization LQ . i.e., LQ [∞]

free-homo

−→

LQ .

(4.8.1)

Proof . By the very definition (4.7.1), LQ [∞]

free-homo

−→

LQ .

Indeed, one can define the embedding map, ∞

∞

n=1

j=−∞

E : LQ [∞] = ⋆ C [{Un }] → ⋆

C [{Uj }] = LQ ,

satisfying E (T ) = T, ∀T ∈ LQ [∞]. Then it is a well-defined injective ∗-homomorphism from LQ [∞] into LQ . Ans, for any free reduced words, s

Ujnl l ∈ LQ [∞],

W = l=1

Certain Banach-Space Operators on LQ [N] ■ 175

where (j1 , ..., js ) ∈ Ns is an alternating s-tuple, satisfying ̸ j3 , ..., js−1 ̸= js , in N, j1 ̸= j2 , j2 = for n1 , ..., ns , s ∈ N, one can get τ (E (W )) = τ (W ) = τ∞ (W ) , in LQ , by Eq. (3.5.1) or by Eq. (1.4.13). It implies that τ (E (T )) = τ∞ (T ) , ∀T ∈ LQ [∞], in LQ , and hence, the ∗-homomorphism E is a free-homomorphism from LQ [∞] to LQ . Therefore, the relation (4.8.1) holds. The above theorem canonically lets us regard LQ [∞] as a free-probabilistic sub-structure of LQ , satisfying the properties of Section 4.7. Differently, one can get the free-isomorphic relation! To consider such a free-isomorphic relation, we define a bijection, g : N → N0 = N ∪ {0} , by

(4.8.2) g(n) = n − 1 in N0 , ∀n ∈ N.

By the existence of the bijection g of Eq. (4.8.2), we naturally obtain the free-isomorphic relation, LQ [∞]

free-iso

=

def ∞

L0Q [∞] = ⋆ C [{Uk+1 }], k=0

(4.8.3)

in LQ . Thus, we identify LQ [∞] with L0Q [∞] by Eq. (4.8.3), and we simply denote L0Q [∞] by LQ [∞] from below, Now, we define a bijection, h : N0 → Z,

176 ■ Constructive Analysis of Semicircular Elements

by the function,

(4.8.4)

def

h(n) =

          

−

0

if n = 0

n 2

if n is even

n+1 2

if n is odd,

in Z, for all n ∈ N0 . Indeed, the function h of Eq. (4.8.4) is a well-defined bijection with its inverse (or, its inverse function), h−1 : Z → N0 , defined by

(4.8.5)

h−1 (j) =

          

0

if j = 0

2j

if j > 0

1 − 2j

if j < 0,

in N0 , for all j ∈ Z. i.e., by Eq. (4.8.5), one has h ◦ h−1 = idZ , and h−1 ◦ h = idN0 , where idY is the identity function on a set Y . Thus, by Eqs. (4.8.2) and (4.8.4), two sets N and Z are equipotent (or bijective) by the existence of the bijection, def

g0 = h ◦ g : N → Z.

(4.8.6)

This bijection g0 of Eq. (4.8.6) induces a bijection G0 : U∞ → U , satisfying

(4.8.7) G0 (Uj ) = Ug0 (j) in U ,

Certain Banach-Space Operators on LQ [N] ■ 177 ∞

∞

for all Uj ∈ U∞ , where U∞ = {Uk+1 }k=0 (resp., U = {Uj }j=−∞ ) is the generating free semicircular family of LQ [∞] (resp., LQ ). It is not hard to verify that the bijection G0 of Eq. (4.8.7) induces a ∗-isomorphism, Φ : LQ [∞] → LQ , satisfying

(4.8.8) Φ (Uj ) = G0 (Uj ) = Ug0 (j) , in U ⊂ LQ ,

for all Uj ∈ U∞ ⊂ LQ [∞]. Theorem 4.17: The semicircular ∞-filterization LQ [∞] is freeisomorphic to the semicircular filterization LQ . i.e., LQ [∞]

free-iso

=

LQ .

(4.8.9)

Proof . As we have seen above, there exists a well-defined ∗isomorphism, Φ : LQ [∞] → LQ , of Eq. (4.8.8). Now, let s

Ujnl l ∈ LQ [∞]

W = l=1

be a free reduced word in U∞ , for an alternating s-tuple (j1 , ..., js ) ∈ Ns , for s ∈ N, and n1 , ..., ns ∈ N. Then τ (Φ (W )) = τ∞ (W ) , by Eqs. (3.5.1) or, (1.4.13) and (4.8.8). (It is not hard to check that the freeness on LQ [∞] is preserved by the freeness on LQ by the ∗isomorphism Φ. i.e., any free reduced words W ∈ LQ [∞] and Φ (W ) ∈ LQ have the same freeness structures, and hence, they induce the same type of noncrossing partitions of (1.4.6)!) It implies that τ (Φ (T )) = τ∞ (T ) , ∀T ∈ LQ [∞],

178 ■ Constructive Analysis of Semicircular Elements

in LQ . Therefore, LQ [∞]

free-iso

=

LQ .

i.e., the free-isomorphic relation (4.8.9) holds true. The above theorem shows that two Banach ∗-probability spaces LQ [∞] and LQ are equivalent as free-probabilistic structures (independent from a canonical free-homomorphic relation (4.8.1)). Observation 4.2: In Section 4.7, under the free-homomorphic relation (4.8.1), the main results of Sections 4.2, 4.3, 4.5, and 4.6 are similarly re-obtained on the semicircular ∞-filterization LQ [∞]. By the free-isomorphic relation (4.8.9) of LQ [∞] and the semicircular filterization LQ , the main results of Section 4.1 can be re-obtained on LQ [∞]. i.e., the integer-shift group B, and the corresponding integer-shift-operator algebra B act on LQ [∞], which is equivalent to LQ , under the inverse Φ−1 of the free-isomorphism Φ of Eq. (4.8.8). And, under the action of Φ−1 , the same results of Section 4.1 are obtained for LQ [∞]. We finish this section with the following result. Theorem 4.18: Let N1 ≤ N2 ≤ ... ≤ ∞ in N∞ = N ∪ {∞}. Then the following free-homomorphic chain holds; LQ [N1 ]

free-homo

−→

LQ [N2 ]

free-homo

−→

···

free-homo

−→

LQ [∞]

free-iso

=

LQ . (4.8.10)

Proof . The free-homomorphic chain (4.8.10) is proven by Eqs. (3.8.6) and (4.8.9). The above theorem characterizes the free-homomorphic relations among the free-probabilistic sub-structures {LQ [N ]}N ∈N∞ in the semicircular filterization LQ by Eq. (4.8.10). Remark that, as we discussed in Remark 3.4, we do not know yet that LQ [N1 ]

free-homo

←−

LQ [N2 ],

or not, where N1 < N2 in N. Extending this question to Eq. (4.8.10), we do not know yet LQ [N ]

free-homo

←−

LQ [∞],

Certain Banach-Space Operators on LQ [N] ■ 179

equivalently, LQ [N ]

free-homo

←−

LQ , ∀N ∈ N,

or not by Eq. (4.8.9).

✥

free-iso

Conjecture 4.1: If N1 ̸= N2 in N∞ , then LQ [N1 ] = LQ [N2 ]. ̸ Suppose N1 ̸= N2 in N, say N1 < N2 in N∞ . Then, by Eq. (4.8.10), we know free-homo LQ [N1 ] −→ LQ [N2 ]. The above conjecture means that, if N1 < N2 in N∞ , then LQ [N2 ] is not free-homomorphic to LQ [N1 ]. If the above conjecture has a positive answer, then the Banach ∗-probability spaces {LQ [N ]}N ∈N are freehomomorphic to the semicircular filterization LQ LQ is not free-homomorphic to them.

free-iso

=

LQ [∞], but

denote

Corollary 4.4: Let BN = (BN , φN ) be a topological ∗-probability N spaces generated by the free semicircular families {xj }j=1 , for all N ∈ N∞ . Assume that the unities 1BN of BN satisfy φN (1BN ) = τN 1LQ [N ] , for all N ∈ N∞ . Then B1

free-homo

−→

B2

free-homo

−→

B3

free-homo

−→

···

free-homo

−→

B∞

free-iso

=

LQ .

Proof . It is proven by Eqs. (3.8.4) and (4.8.10).

REFERENCES [1] I. Cho, Banach-Space Operators Acting on Semicircular Elements Induced by Orthogonal Projections, Compl. Anal. Oper. Theory, 13, no 8, (2019) 4065–4115. [2] I. Cho, Banach-Space Operators Acting on Semicircular Elements Induced by p-Adic Number Fields over Primes p, Conference Proceeding of RAiMS (July 2019), Mathematics, 7, (2019) 498. [3] I. Cho, Acting Semicircular Elements Induced by Orthogonal Projections on von Neumann Algebras, Mathematics, 5, (2017) 74.

180 ■ Constructive Analysis of Semicircular Elements [4] I. Cho, A Certain Group Acting on C ∗ -Algebras Generated by Countably Many Semicircular Elements, Conference Proceeding of ICMRS 2019: Mathematical Models and Methods in Applied Sciences, Antalya, Turkey, Springer, (2019). [5] I. Cho, Certain Banach-Space Operators Acting on Free Poisson Elements Induced by Orthogonal Projections. In: H. Dutta (ed.), Topics in Contemporary Mathematical Analysis and Applications. Taylor & Francis: New York, 1–48 (2020). [6] I. Cho, Asymptotic Semicircular Laws Induced by p-Adic Number Fields over Primes, Compl. Anal. Oper. Theory, 13, no. 7, (2019) 3169–3206. [7] I. Cho, Free Stochastic Integrals for Weighted-Semicircular Motion Induced by Orthogonal Projections. In: H. Dutta (ed.), Applied Mathematical Analysis: Theory, Methods, and Applications. Taylor & Francis: New York, 371–402 (2020). [8] I. Cho, Acting Semicircular Elements Induced by Orthogonal Projections on von Neumann Algebras, Mathematics, 5, (2017) 74. DOI: 10.3390/math5040074. [9] I. Cho, and J. Dong, Catalan Numbers and Free Distributions of Mutually Free Multi Semicircular Elements, Compl. Anal. Oper. Theory, (2020). Submitted to Compl. Anal. Oper. Theo. [10] I. Cho, and P. E. T. Jorgensen, Semicircular Elements Induced by Projections on Separable Hilbert Spaces. In: D. Alpay, and M. Vajiac (eds.), Operator Theory Advances & Applications. Birkhauser: Cham (2018). DOI: 10.1007/978-3-030-18484-1-6. [11] I. Cho, and P. E. T. Jorgensen, Banach *-Algebras Generated by Semicircular Elements Induced by Certain Orthogonal Projections, Opuscula Math., 38, no. 4, (2018) 501–535. [12] I. Cho, and P. E. T. Jorgensen, Free-Homomorphic Relations Induced by Certain Free Semicircular Families, In: Monograph in Honor of Prof. Victor Katsnelson. Springer: New York (2020). [13] A. Connes, Noncommutative Geometry. Academic Press: San Diego, CA (1994). ISBN: 0-12-185860-X. [14] P. R. Halmos, Graduate Texts in Mathematics: Hilbert Space Problem Books. Springer (1982). ISBN: 978-0387906850.

Certain Banach-Space Operators on LQ [N] ■ 181 [15] A. Nica, and R. Speicher, Lectures on the Combinatorics of Free Probability, (1st Ed.). London Mathematical Society Lecture Note Series, 335. Cambridge University Press: Cambridge (2006). ISBN-13:9780521858526. [16] I. Nourdin, G. Peccati, and R. Speicher, Multi-Dimensional Semicircular Limits on the Free Wigner Chaos, Progr. Probab., 67, (2013) 211–221. [17] F. Radulescu, Random Matrices, Amalgamated Free Products and Subfactors of the C*-Algebra of a Free Group of Nonsingular Index, Invent. Math., 115, (1994) 347–389. [18] F. Radulescu, Free Group Factors and Hecke Operators, notes taken by N. Ozawa, Proceeding of 24th Conference in Operator Theory, Theta Advanced Series in Mathematical, Theta Foundation, Univ. of the West Timisoara, Romania, 2012. (2014). [19] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Am. Math. Soc. Mem., 132, no. 627, (1998) x+88. [20] R. Speicher A Conceptual Proof of a Basic Result in the Combinatorial Approach to Freeness, Infinit. Dimention. Anal. Quant. Prob. & Related Topics, 3, (2000) 213–222. [21] R. Speicher, and T. Kemp, Strong Haagerup Inequalities for Free RDiagonal Elements, J. Funct. Anal., 251, no. 1, (2007) 141–173. [22] R. Speicher, and U. Haagerup, Brown’s Spectrial Distribution Measure for R- Diagonal Elements in Finite Von Neumann Algebras, J. Funct. Anal., 176, no. 2, (2000) 331–367. [23] D. Voiculescu, Aspects of Free Analysis, Jpn. J. Math., 3, no. 2, (2008) 163–183. [24] D. Voiculescu, Free Probability and the Von Neumann Algebras of Free Groups, Rep. Math. Phy., 55, no. 1, (2005) 127–133. [25] D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, vol. 1. American Mathematical Society: Ann Arbor, MI (1992). ISBN-13: 978-0821811405.

CHAPTER

5

Discussion

n this section, we briefly summarize our main results of Chapters 2–4. Motivated by the constructions and main results of Chapter 2 (Also, see e.g., [5,6]), we established mutually free semicircular elements U = {Uj }j∈Z from mutually orthogonal |Z|-many projections and studied the semicircular filterization LQ generated by U in Chapter 3. The free-probabilistic sub-structures {LQ [N ]}N ∈N∞ of LQ , generated by the free semicircular (sub-)families {UN }N ∈N∞ of U , are studied in Chapters 3 and 4. Especially, in Chapter 4, we considered Banach-space operators generated by certain ∗-homomorphisms acting on our Banach ∗-probability spaces, preserving-or-distorting the original free-distributional data. It is realized that those operators act like the classical multiplication or Toeplitz (Hilbert-space) operators (e.g., [1,7–9]), and hence, the deformations of free-probabilistic information are related to such actions. We emphasize at this moment that our main results can be generalized or universalized in more abstract settings (e.g., see Refs. [3,4]). The main results of Chapter 2 not only provide interesting freeprobabilistic structures in quantum statistical physics but also gives motivations for Chapters 3 and 4; how to construct (weighted-) semicircular elements from fixed mutually orthogonal projections with help of analysis on p-adic number fields Qp . We showed how to construct weighted-semicircular elements from the analyses, and obtain corresponding free-distributional data in various different formats. In fact, weighted-semicircular elements are more interesting than the semicircular elements under the number-theoretic settings of Chapter 2

I

DOI: 10.1201/9781003374817-5

183

184 ■ Constructive Analysis of Semicircular Elements

because of the universality of the semicircular law. That is, the weighted-semicircular elements of Chapter 2 contain number-theoretic information came from primes, even though the constructions of semicircular elements from such number-theoretic objects are one of the main results of the section. The main purpose of Chapter 3 is to construct semicircular elements from fixed mutually orthogonal |Z|-many projections, and the provided way is motivated by that of Chapter 2 different from the earlier works. As application, we generalize or abstractize the main results of Chapter 2. i.e., whenever there are orthogonal countable-infinitely many projections, one can have mutually free, semicircular elements in so-called the free filterization induced by the projections. It gives a direct connection between operator theory and free probability theory. The main results of Chapter 4 show that our (weighted-) semicircularity is preserved-or-distorted by certain actions on the free filterization. Such deformations are characterized under some natural conditions. To do that, we studied certain Banach-space operators acting on free probability spaces containing our semicircular elements. Interesting properties of those operators are observed therein. As we mentioned in the above text, there are interesting-andimportant open problems left. In particular, the “complete” freehomomorphic relations among {LQ [N ] : 1 ≤ N ≤ ∞} ∪ {LQ } are needed to be solved for its own importance and for its applications to other fields (e.g., [2]). Are LQ [2] and LQ [3] free-isomorphic?

REFERENCES [1] A. Bottcher, and B. Silbermann, Analysis of Toeplitz Operators. SpringerVerlag: New York, Berlin Heidelberg (1990). ISBN: 0-387-52147-X. [2] I. Cho, Algebras, Graphs and Their Applications. CRC Press, Taylor & Francis Group: New York (2014). ISBN:978-1-4665-9019-9. [3] I. Cho, Discrete Dynamics on C ∗ -Probability Spaces Generated by Multi Semicircular Elements, Springer: New York (2021). [4] I. Cho, Certain Banach-Space Operators on C ∗ -Probability Spaces Generated by Multi Semicircular Elements. Springer: New York (2020).

Discussion ■ 185 [5] I. Cho, Asymptotic Semicircular Laws Induced by p-Adic Number Fields over Prime p, Compl. Anal. Oper. Theory, 13, no. 7, (2019) 3169–3206. [6] I. Cho, and P. E. T. Jorgensen, Deformations of Semicircular and Circular Laws via p-Adic Number Fields and Samplings of Primes, Opuscula Math., 39, no. 6, (2009) 771–811. [7] J. B. Conway, A Course in Functional Analysis. Springer-Verlag: New York, Berlin Heidelberg, Tokyo (1985). ISBN: 0-387-96042-2. [8] J. B. Conway, Subnormal Operators. Pitman Books Limited: London (1981). ISBN: 0-273-08520-4. [9] P. R. Halmos, A Hilbert Space Problem Book, (2nd Ed.). Springer-Verlag: New York, Heidelberg Berlin (1982). ISBN: 0-387-90685-1

Index adjoint 4 annihilation 92 automorphism group 116 Banach Banach Banach Banach

field 36 space 36 ∗-algebra 3 ∗-probability space 3

Catalan number 4 circular element 10, 67, 106, 164 C ∗ -algebra 3 conditional nilpotent operator 160 ∗ C -probability space 3 creation 92 crossed product 117 crossed product topological ∗-algebra 117 dynamical semicircular ∗-probability space 119 filter 92 filterization 98 free 3 free Adelic filterization 59 free cumulant 3 joint free cumulant 3 mixed free cumulant 3 free cumulant sequence 4 free distribution 4 free filterization 99 free-homomorphism 112

free-isomorphism 112 free moment 3 joint free moment 3 free moment sequence 4 free Poisson element 11, 68, 106, 169 free probability 3 free random variable 3 free semicircular family 64 free weighted-semicircular family 64 group-dynamical system 117 group-dynamical ∗-probability space 117 Hilbert space 44 Hilbert-space representation 45 homomorphism semigroup 150 integer shift 116 integer-shift group 116 Kronecker delta 5 linear functional 3 bounded linear functional 3 measurable subset 36 measure space 36 Moebius inversion 5 p-adic filter 58 p-adic number field 35 p-annihilation 53 187

188 ■ Index

p-boundary subalgebra 50 p-C ∗ -algebra 46 p-C ∗ -measure space 48 p-creation 53 p-norm 35 p-radial operator 53 prime 35 prime-integer shift (pi-shift) 74 projection 87 orthogonal projections 87

semicircular element 4, 60 semicircular filterization 101 semicircular ∞-filterization 170 semicircular N -filterization 133 shift operator (integer-shift operator) 142 (restricted-)shift operator 153 sigma algebra (σ-algebra) 36 star algebra (∗-algebra) 44

radial operator 92 radial-projection algebra 93 representation 45 restricted-shift family 152

von Neumann algebra 3

self-adjoint 4 semicircular Adelic filterization 66

weighted semicircular element 60 weighted semicircular laws 60 W ∗ -probability space 3